Combination and permutation formula. Also, read: Permutation and combination.

Aug 17, 2021 · Combinations. Permutation without Repetition: This method is used when we are asked to reduce 1 from the previous term for each time. For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple Jun 26, 2020 · How to calculate combination and permutation in C - Combination and permutation are a part of Combinatorics. More abstractly, each of the following is a permutation of the letters \ ( a, b, c,\) and \ (d\): Note that all of the objects must appear in a permutation and two orderings are Sep 17, 2023 · The Combinations Calculator will find the number of possible combinations that can be obtained by taking a sample of items from a larger set. A club has 10 members and needs to choose a president and vice president. If we take into consideration the order in which we select a fruit first and then after the second one gives a situation of permutation. If 'n' represents the total number of things and 'r' is less than or equal to n, then. In the case that n is 12 and r is 2, find the total number of permutations and combinations that can happen. There are 4 different colors to paint a car. In combinatorics, a permutation is an ordering of a list of objects. For this you would want to use combinations not permutations, since the dice are indistinguishable. P (10,3) = 720. r = 3. 1. For selecting the team members, choosing food menu, drawing lottery and so on. Permutations and One could say that a permutation is an ordered combination. A permutation of some objects is a particular linear ordering of the objects; P(n, k) in effect counts two things simultaneously: the number of ways to choose and order k out of n objects. (r + n -1)! r! × (n - 1)! This free calculator can compute the number of possible permutations and Jan 6, 2024 · A permutation is the number of ways a particular data set or sample can be arranged or rearranged. =PERMUT (3,2) Permutations possible for a group of 3 objects where 2 are chosen. A code have 4 digits in a specific order, the digits are between 0-9. Solved Problems Based on the Permutation and Combination Formulas. 6. 2. Permutation = n P r = n!/ (n-r)! Combination = n C r = n P r /r! where, n, r are non negative integers and r ≤ n. The number of ordered arrangements of r objects taken from n unlike objects is: n P r = n! . Hundreds place: 5 choices . This generalises to other combinations too and gives us the formula #combinations = n! / ((n - r)! * r!) Explaining permutations with repetitions formula. 0! To use PERMUT, specify the total number of items and " number_chosen ", which represents the number of items in each combination. Some really tricky problems can offer up a mixture of the two. Pn = (n – 1) ! ⇒ Pn = (5 – 1) ! ⇒ Pn = 4! Step 3: Calculate 4! (4 factorial): Pn = 4 Mar 8, 2023 · For PERMUTATIONS, order matters. Enter the total number of objects (n) and the number of elements taken at a time (r) 3. In combinatorics, the combination is a way of selecting something from a given collection. 375) The formula for combination helps to find the number of possible combinations that can be obtained by taking a subset of items from a larger set. When we encounter n! (known. we use combinations formulas. There are 10 possible cars to finish first. For example, let’s take the set {A, B, C}. This is a permutation of 10 items taking three at a time. In permutations, order/sequence of arrangement is considered, unlike in combinations. The number of permutations of n objects taken r at a time is determined by the following formula: P(n, r) = n! (n − r)! n! is read n factorial and means all numbers from 1 to n multiplied e. 1) (7. For this calculator, the order of the items chosen in the subset does not matter. Using the k objects in c, we can create a total of kPk = k! unique k -permutations that each contains exactly the k objects. What are Permutation and Combination Formulas? Here are the permutations and combinations formulas. where, n is the size of the set from which elements are permuted; r is the size of each permutation! is factorial operator; Relation between Combinations Apr 23, 2020 · Thus, to eliminate the unnecessary groups that are the same, you divide the number of original 6,720 Permutations by 5!. Combination is is the different ways of selecting elements if the elements are taken one at a t Formula for permutation: nPr = n!/(n-r)! Formula for combination: nCr = n!/r!. Select whether you would like to calculate the number of combinations or the number of permutations using the simple drop-down menu. Select whether repeat elements are permitted. 970200. However, most of these permutation combination formulas are based on two essential formulas. In the end you can test your knowledge based on permutation and combination questions for Class 11. The general permutation can be thought of in two ways: who ends up seated in each chair, or which chair each person chooses to sit in. Assume that 10 cars are in a race. There are \(\binom{52}{13}\) ways to give 13 cards to the first player. The number of permutations of n objects taken r at a time is determined by the following formula: P(n, r) = n! (n − r)! Example. Explained separately in a more accessible way: Combination. The most important formulas are: nPr = n! / (n−r)! nCr = n! / (r! * (n−r)!) nPr = n! / (n−k)! In mathematics, permutation refers to the arrangement of all the members of a set in some order or sequence, while combination does not regard the order as a parameter. 1) 7 C 3 = 7 P 3 3! = 7! 4! ∗ 3! In a combination in which the order is not important and there are no assigned roles the number of possibilities is defined as: Jun 4, 2021 · by Zach Bobbitt June 4, 2021. Unlike permutations, the order in which the items are selected does not matter. Combinations sound simpler than permutations, and they are. Proof of Lemma 1: Our goal is to show that exactly k! ⋅ nCk unique k -permutations can be created from the nCk k -combinations. Jul 11, 2023 · In other words, permutations consider the different ways in which objects can be arranged. Step 2: Put those letters in order. The formula of permutation and combination can be combined to form a single formula. A useful special case is k = n, in which we are simply counting the number of ways to order all n objects. 5 Permutations possible for the arguments specified in A2:A3. Actually, these are the hardest to explain, so we will come back to this later. At first glance, counting seems trivial What are permutations and combinations? A permutation is an arrangement in a definite order of a number of objects taken, some or all at a time. Jun 26, 2020 · Combination is is the different ways of selecting elements if the elements are taken one at a time, some at a time or all at a time. Example 1: Find the number of permutations and combinations: n =6; r = 4. This result can be seen in cell D8 in the example shown. Permutations A permutation is an arrangement of objects in which order is important. In how many ways the letters of the word ‘CLEVER’ can be arranged? a) 240. In Section 2. Permutation can be done in two ways, Permutation with repetition: This method is used when we are asked to make different choices each time and with different objects. Don’t memorize the formulas, understand why they work. Combinations. Aug 16, 2023 · Use the multiplication principle. See also. Once a car has finished first, there are nine cars to finish second. Some important This is a problem that combines permutations and combinations. For example, there are 6 permutations of the letters a, b, c: . Using the nPr notation, from a set of 3 objects we are choosing 3. This is usually written n P k . May 3, 2023 · The Permutation formula. It follows a particular order or sequence. This is less important when the two groups are the same size, but much more important when one is limited. c) 120. Jan 23, 2022 · Given an array arr[] of distinct elements and two integers M and K, the task is to generate an array from the given array elements (elements can repeat in the generated array) such that the size of the generated array is M and the length of any sub-array with all same elements must not exceed K. One could say that a permutation is an ordered combination. A combination May 30, 2024 · Learn the concepts of permutation and combination, and how to use formulas to calculate them. Combinations with Repetition. Use the multiplication principle. Halfway through that explanation, you might have realized we were recreating the combination formula: That's the shortcut when you know order doesn't matter. This topic is not only important for school studies, but it is also important for other competitive examinations. The result is 6,720 / 120 = 56. 4 days ago · Concept of Permutation and Combination. Q. There are two permutation formulas. However, be careful! Therefore, this is a problem in combinations. 10. In combinatorial counting, the order is irrelevant. Returns the number of permutations for a given number of objects that can be selected from number objects. After the second car is finished, any of the eight remaining cars can finish third. , $6720$ by $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ to get $6720 /5 = 56$ combinations. Again for the curious, the equation for combinations with replacement is provided below: n C r =. 1 = 330 ways. For repetitions, the formula is: n r. Therefore, only the first row is relevant, i. For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Mar 10, 2024 · Combination generator. The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. When we speak of counting, it is shorthand for determining the size of a set, or more often, the sizes of many sets, all with something in common, but different sizes depending on one or more parameters. Factorials. Aptitude Questions on Permutation and Combination involve a lot of formulas. There are also two types of combinations (remember the order does not matter now): Repetition is Allowed: such as coins in your pocket (5,5,5,10,10) No Repetition: such as lottery numbers (2,14,15,27,30,33) 1. Learn about permutation and combination formula. We know that we have them all listed above —there are 3 choices for which letter we put first, then 2 choices for which letter comes next, which leaves only 1 choice for the last letter. b) 720. The general formula is therefore # Combinations and permutations in the mathematical sense are described in several articles. 3. 5. different lineups. 7. Solution. , there are 4 possible combinations. In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations ). k particular objects are always included = n-k C r-k. C is the symbol for combination. Use the formula for the number of permutations. Example: A farmer purchased 3 cows, 2 pigs, and 4 hens from a man who has 6 cows, 5 pigs, and 8 hens. The formula for 'r' things taken from 'n' things is such that their permutation is equal to the product of 'r' factorial and combination. In some scenarios, the order of outcomes matters. Permutation = factorial(n) / factorial(n-r); Combination = factorial(n) / (factorial(r) * factorial(n-r)); n = 5. Learn how to recognise between a permutation case and a combination case. Consider the above example of selecting two fruits from the four. Nov 30, 2023 · Step 1: Understand the circular permutations formula: Pn = (n – 1) ! This formula computes the number of configurations in a circle when rotations are considered separate. We don't mean it like a combination lock (where the order would definitely matter). We’ll do this in two steps: Step 1: Choose 3 letters (paying no attention to order). . Jul 5, 2024 · Combinations Formula. A permutation involves the selection of 'r' items from a set of 'n' items, where the order of selection matters and replacement is not allowed. 🔗. An example of this is given as follows −. It contains a few word problems including one associated with the fundamental counting princip Feb 17, 2024 · Combination is a technique for determining the number of different possible arrangements where the order of selection is not relevant. It is of paramount importance to keep this fundamental rule in mind. The permutation of two things from three given things \ (p, q, r\) is \ (pq, qp, qr, rp, pr, rp. For example, if you have a lock where you need to Jun 11, 2024 · Combination Formula. Typically, this method is used for large data sets; the researcher need not record each possibility separately. k particular objects are never included = n-k C r. While this sounds simple, perhaps too simple to study, it is not. 1 we investigated the most basic concept in combinatorics, namely, the rule of products. Use the formula for the number of combinations. Learn how to calculate permutations and combinations of objects using various formulas and examples. We have discussed a few permutations and combinations formulas questions below: Problem 1. (n-r)! Difference between permutation and combination *In permutations, the order matters*, so rearranging the order of selected objects results in different permutations. Factorial. The formula is n C r = n ! / r ! * ( n – r )! Here n denotes the total numerical value and r denotes the number of probabilities we are considering. Step 2: Apply the formula to the problem: We need to place 5 balls in a circle. C (10,3) = 120. Basically, it shows how many different possible subsets can be made from the larger set. We turn first to counting. Described together, in-depth: Twelvefold way. (Permutation and Combination class 11) Here we have three vacant places, where digits have to be placed according to given conditions. r is the size of each permutation. Learn Permutation and combination in detail with real life examples. 2. Therefore, to evaluate the total number of combinations, we divide the total permutations, i. n C r = Number of combinations (selections) of n things taken r at a time. Example 2. These questions are beneficial for various competitive exams, placement interviews, and entrance tests. It shows how many different possible subsets can be made from the larger set. How many unique color combinations are there if order matters (e. Choosing a subset of r elements from a set of n elements; and 2. The two fundamental formulas are: Permutation Formula. Finding the Number of Permutations of n Distinct Objects Using a Formula. \) The combination of two things from three given things \ (p, q, r\) is \ (pq Sep 29, 2021 · A permutation is a (possible) rearrangement of objects. Formulas for Permutations Oct 6, 2021 · In order to find the actual number of choices we take the number of possible permutations and divide by 6 to arrive at the actual answer: 7C3 = 7P3 3! = 7! 4! ∗ 3! (7. In combination, we can determine different ways of selecting 4 persons from 10 persons. The number of ways in which r things at a time can be SELECTED from from n things is Combinations (represented by n C r ). We know that we have them all listed above —there are 3 choices for which letter we put first, then 2 choices for which letter comes next, which leaves only 1 choice for the 3 days ago · This is the key permutation combination difference that you should understand to consolidate the concept. If we again picked 3 out of 5 balls but with repetitions then we have 5 options for each selection, giving us 5 * 5 * 5 = 125 selections overall. Find out the difference between permutation and combination, and how to derive the formulas using factorials and counting principles. Permutation. Take any combination c from C. For example, to calculate 3-number permutations for the numbers 0-9, there are 10 numbers and 3 chosen, so the formula is: = PERMUT (10,3) // returns 720. ABC ACB BAC BCA CAB CBA Combinations. Combination formula is used to pick r things out of n different things, where the order of picking is not important, and replacement is not allowed. Jan 25, 2023 · 1. This is special because there are no positive numbers less than zero and we Also, try important permutation and combination questions for class 11. 3. Get Permutation and Combination Class 11 NCERT Solutions for on Embibe. If it is given that n is 12 and r is 2, and if we use the formula We don't mean it like a combination lock (where the order would definitely matter). Permutation = 60. See examples, solved problems, and FAQs on this topic. So, we can conclude that The number of permutations of n different things taken r at a time, when m particular things are always to be included = n-m C r-m × r! 6. A factorial is represented by the sign (!). Probability of "at least 2 heads in a row" is 3/8th (0. Also after selecting the fruits if we arrange them in an order the above situation of Permutations: The order of outcomes matters. 3 P 3! 3 = = = = 3! 3! 3! • In our list of 210 sets of 3 professors, with order mattering, each set of three profs is counted 3! = 6 times. This section develops formulas for both permutations and combinations. Permutation Formula: A permutation is the arrangements of r things from a set of n things without replacement Formulas for Combinations. Permutations formula is, n P r = (n!) / (n - r)!. Also, examples of both permutation and combination for class 11 are given for students’ reference. So, it is mandatory to get the information on Maths class 11 tricks, permutation and combination formulas also. Permutation refers to the arrangement, and combination refers to selection. ! is the factorial operator. 6 above, Gomer is choosing and arranging a subset of 9 Combinations. This is n(n − 1)⋯(n − n + 1) = n!. Perhaps a better metaphor is a combination of flavors — you just need to decide which flavors to combine, not the order in which to combine them. To generalize, in order to arrive at the number of Combinations, you need to figure out all the Permutations and divide by all the Redundancies. 8)/4. Permutation is the different arrangements that a set of elements can make if the elements are taken one at a time, some at a time or all at a time. Example 1. Also, read: Permutation and combination. , red-blue-green is different from green-blue-red)? 2. The only constant thing is that n should always be May 22, 2024 · There are several formulas associated with the concepts of permutation and combination. Let’s understand this difference between permutation vs combination in greater detail. A permutation is an ordered arrangement. a b c, a c b, b a c, b c a, c a b, c b a. n and r are dictated by the limiting factor in question: which people get to be seated in each of the limited number of chairs (n = # of people, r = # of 2! × 9! = 55. Permutations and Combinations. In this section we Jun 4, 2024 · Definition, Formula, Permutations and Combinations; Practice Problems on Permutation Formula. It differs from a combination; multiple scenarios can be determined from a single combination. Eg. Referring to EXAMPLE 1. Specifically, for a selection of items to May 20, 2024 · Permutation is similar to the combination and the basic difference between permutation and combination is that in permutation the order in which the object is taken is important while the combination is the arrangement of the objects when the order of the objects is not important. be/2ctYfLvmHHU Dec 28, 2015 · Lemma 1: nPk ≥ k! ⋅ nCk. Relation Between Permutation and Combination Formula. The number of arrangement of n distinct objects taken r This video tutorial focuses on permutations and combinations. (n – r)! Example. A permutation of n distinct objects taken r at a time is formed by filling of r positions, in a row with objects The relationship between combinations and permutations is n C r = n P r /r! The number of ways of selecting r objects from n different objects subject to certain condition like: 1. choose(n, r) #calculate total permutations of size r from n total objects choose(n, r) * factorial(r) The following examples show how to use each of these functions in Calculate Combinations and Permutations in Five Easy Steps: 1. (For k = n, nPk = n! Thus, for 5 objects there are 5! = 120 arrangements. There are 3 ∗ 2 ∗ 1 = 3! arrangements of 3 objects. Permutation and combination have a lot of similarities but they also have some striking differences. ) For combinations, k objects are selected from a set of n objects to produce subsets without ordering. However, sometimes I'm not sure whether I need a permutation or combination from the outset. Approach 2: Just Use the Combination Formula. Using the permutation formula: Learn the difference between permutations and combinations, using the example of seating six people in three chairs. It includes 2 wicketkeepers and 5 bowlers. We’ve already seen how to compute the number of permutations using the formula To compute the number of combinations, let’s count them another way using the Multiplication Rule for Counting. A Combination is a selection of some or all, objects from a set of given objects, where the order of the objects does not matter. For instance, the 6 possible permutations of the letters A, B, and C are shown. May 31, 2024 · For example, using this formula, the number of permutations of five objects taken two at a time is. Fortunately, we can solve these problems using a formula. Jan 10, 2019 · A permutation is a (possible) rearrangement of objects. The number of permutations can be calculated using the formula: nPr = n! / (n – r)! The PERMUTATION FORMULA The number of permutations of n objects taken r at a time:! P(n,r)= n! (n"r)! This formula is used when a counting problem involves both: 1. Before we learn the formula, let’s look at two common notations for permutations. Formulas. In how many ways can you select a cricket team of eleven players if you have to This set of Aptitude Questions and Answers (MCQs) focuses on “Permutations and Combinations”. 5! = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1. As we had suggested earlier, the best approach is to start from scratch, using the addition and/or multiplication principles, along with permutation and/or combination whenever it seems appropriate. In how many ways can three cars finish in first, second and third place? The order in which the cars finish is important. Basic Formula of Permutation and Combination. Ans: n = 15, r = 3 (Given) Using the formulas for permutation and combination, we get: Permutation, P = n!/(n – r)! Apr 20, 2015 · Permutation with Repetition is the simplest of them all: N to the power of R. It makes sense that there are fewer choices for a combination than a permutation, since the redundancies are being removed. $6^3$ gives the number of possible rolls under the assumption that $(3,3,2)$ is different than $(3,2,3)$ but since this is a casino and casinos don't have numbered or different colored die, $(3,3,2)$ should be the same as $(3,2,3)$ since there is no way to tell these two results are distinct. IGCSE Add Maths - Permutations & Combinations. What is the Permutation Formula, Examples of Permutation Word Problems involving n things taken r at a time, How to solve Permutation Problems with Repeated Symbols, How to solve Permutation Problems with restrictions or special conditions, items together or not together or are restricted to the ends, how to differentiate between permutations and combinations, with video lessons, examples and May 26, 2022 · Let’s look at a simple example to understand the formula for the number of permutations of a set of objects. You can use the following functions to calculate combinations and permutations in R: #calculate total combinations of size r from n total objects. Arranging the chosen elements. Free, unlimited, online practice. g. Permutation: Listing your 3 favorite desserts, in order, from a menu of 10. N is the number of things you are choosing from, r is the number of items. Permutations. For example, arranging four people in a line is equivalent to finding permutations of four objects. 2 Combinations and permutations. The number of combinations of n objects, taken r at a time represented by n Cr or C (n, r). as ‘n factorial’) we say that a factorial is the product of all the whole numbers. The combinations formula provides a way to calculate the number of combinations of n different things taken r at a time is given by. For example, let’s say you are choosing 3 numbers for a combination lock that has 10 numbers (0 to 9). Number of Circular Permutations. In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Permutations and To work out the solutions of probability mathematics with the use of the combination, factorial is used. e. Nov 21, 2023 · Combinations, along with permutations, form the foundation of an area of discrete mathematics known as combinatorics. It is just a way of selecting items from a set or collection. We know that we have them all listed above —there are 3 choices for which letter we put first, then 2 choices for which letter comes next, which leaves Jan 11, 2012 · Here’s the permutation formula: # of permutations of n objects = n! So, five books the number of permutations is 5! = (5)(4)(3)(2)(1) = 120. Formula: Example: How many ways can 4 students from a group of 15 be lined up for a photograph? Answer: There are 15 P 4 possible permutations of 4 students from a group of 15. For example, we have to form a team of 4 people from the given ten persons. For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. A permutation is a (possible) rearrangement of objects. 10 C 3 = 10! = 10 × 9 × 8 = 120 3! (10 – 3)!3 × 2 × 1. n is the size of the set from which elements are permuted. Which one you choose depends on whether you have repetitions. 9. Many permutation and combination formula aptitudes are there in Mathematics. n P r = (n!) / (n-r)! Combination Formula. ”. 1: Find the number of permutations and combinations, if n = 15 and r = 3. between 1 and n, where n must always be positive. It should be noted that the formula for permutation and combination are interrelated and are mentioned below. For COMBINATIONS, order doesn't matter. Suppose there is a class of 20, and we are going to pick a team of three people at random, and we want to know: how many different possible Sep 27, 2022 · #combination #permutation #msword How to write Matrix of order greater than 3x3https://youtu. This is read five factorial. In these, "at-least-2 Heads in a row" permutations are: HHH, HHT, THH - 3. The permutations of this set would be ABC, ACB, BAC, BCA, CAB, and CBA. *In combinations, the order does not matter*, so different arrangements of the same set of Nov 21, 2023 · A combination is a way of selecting certain items within a group of items. A permutation is any set or subset of objects or events where internal order is A permutation is a (possible) rearrangement of objects. For example. It also contains meaning and definition with solved examples. Permutation is represented by the letter, P. A combinationis a selection from a larger set. Example: 3 tosses of 2-sided coin is 2 to power of 3 or 8 Permutations possible. We can do this by using the combination formula as: 11 C 4 = 11!/4! (11-4)! = 11!/7! = (11. To further illustrate the connection between combinations and permutations, we close with an example. A formula for the number of possible permutations of k objects from a set of n. Use combinations and the Binomial Theorem to expand binomials. Print the maximum sum of the elements among all the p Combination: Choosing 3 desserts from a menu of 10. For example, there are 6 permutations of the letters a, b, c: abc, acb, bac, bca, cab, cba. . This combination calculator (n choose k calculator) is a tool that helps you not only determine the number of combinations in a set (often denoted as nCr), but it also shows you every single possible combination (or permutation) of your set, up to the length of 20 elements. Scroll down the page for examples and solutions on how to use the formulas to solve examination word problems. Permutations count the different arrangements of people in specific chairs, while combinations count the different groups of people, regardless of order or chair. For n different objects, we have to make r Sep 2, 2019 · Combination questions will indicate that you need to form groups or sets; permutation questions will have words or phrases that indicate order, such as “first, second, third” or “how many different ways. What are the combinations of letters A, B, C, and D in a group of 3? There are 24 possible permutations when the order is important. The following figure gives the formula for Permutations and Combinations. This is exactly what the Combinations Calculator calculates. Tens place: 5 Jun 21, 2023 · We have provided some permutation and combination examples with detailed solutions. The permutation formula is as follows: If n, r are positive integers and r ≤ n, then the number of permutations of n distinct objects taken r at a time is n (n − 1) (n − 2) · · · (n − r + 1). ^n P_r = \frac {n!} { (r)! (n-r)!} ; where n ≥ r (n is greater than or equal to r). The number of combinations of n different things taken r at a time is denoted by n C r or C (n, r) ⇒ n C r = \(\frac{n !}{r !(n-r) !}\) 8. As the old adage says, “practice makes perfect”—the more of Combination. n C r = n! ⁄ r! (n-r)! ,0 < r ≤n. The number of distinct combinations of 3 professors is. Permutation and the combination is the study of arranging elements in a set, and of combining and rearranging elements. Permutations may be represented by a specific sequence of permutation numbers, while combinations may be represented by a specific sequence of combination numbers. 0! is a special case factorial. And then you’ll learn how to calculate the total number of each. Combinatorics is all about counting. Nov 1, 2023 · Some Permutations and Combinations Examples… Que 1: How many three-digit numbers can be formed with the digit 1,2,3,4,5? Case 1: The repetition of digits is allowed. Example 2: The Indian Cricket team consists of 16 players. 10 x 9 x 8 = 720. Permutation Formulas. Combinations: The order does not matter. 2 we saw a subclass of rule-of-products problems, permutations, and we derived a formula as a computational aid to assist us. bs sg lw we su so df jo vt wq