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# Permutation And Combination

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### Permutation And Combination

1. 1. Permutation and Combination Probability -1
2. 2. Fundamental rule of Counting <ul><li>If one operation can be performed in m different ways and another operation can be performed in n different ways ,then the two operations when associated together can be performed in m*n ways </li></ul>
3. 3. Factorial of n <ul><li>The Continued product of first n natural Numbers is called factorial n and is denoted by n! </li></ul>
4. 4. Permutation <ul><li>illustrates the number of ways to arrange elements in a definite order. The number of permutations of n elements of same kind taken r at a time. </li></ul>
5. 5. Example <ul><li>Write down all the permutations of xyz. </li></ul><ul><li>xyz, xzy, yxz, yzx, zxy, zyx. </li></ul>
6. 6. Example <ul><li>In how many ways can 3 of 8 different flowers be planted along one side of the road? nP r = 8P 3 = 336 </li></ul>
7. 7. Example <ul><li>.    Express  10 P 4  in terms of factorials. </li></ul><ul><li>    Solution .   10 P 4  =   10!  6! </li></ul>
8. 8. Combination <ul><li>Combination illustrates the number of ways to arrange elements without a definite order. The number of combination of n elements taken r at a time. </li></ul>
9. 9. Example <ul><li>In how many ways can a committee of 4 be selected from a group of 12 people? nC r = 12C 4 = 495 </li></ul>
10. 10. <ul><li>The number of combinations of 4 things taken 3 at a time. </li></ul><ul><li>We will denote this number as  4 C 3 .  In general, </li></ul>
11. 11. Problem <ul><li> 1. Write all the combinations of abcd taken 1 at a time. </li></ul><ul><li>2.Write all the combinations of abcd taken 2 at a time. </li></ul><ul><li>3. Write their combinations taken 3 at a time. </li></ul><ul><li>4. Write their combinations taken 4 at a time. </li></ul>
12. 12. Question No 1 <ul><li>What is the probability of getting 3 white balls in a draw of of 3 balls from a box contain 5 white and 4 black balls. </li></ul>
13. 13. Question no 2 <ul><li>A committee is to be constituted by selecting 3 people from a group consisting of 3 economists and 4 statistics find the probability that the committee consists of </li></ul><ul><li>Economists only </li></ul><ul><li>Two scientists </li></ul><ul><li>One economists and one statistician </li></ul>
14. 14. Question no 3 <ul><li>A sub committee of 6 members is to be formed out of group consisting of 7 men and 4 ladies. Obtain the probability that the sub committee will consist of </li></ul><ul><ul><ul><li>Exactly two ladies </li></ul></ul></ul><ul><ul><ul><li>Atleast 2 ladies </li></ul></ul></ul>
15. 15. Question no 4 <ul><li>A committee of 4people is to be appointed from 3 officers of the production department for offices of purchase department ,2 officers of sales department and one chartered accountant .find the probability of forming the committee in the following manner. </li></ul><ul><li>There must be one form each categories </li></ul><ul><li>It should have atleast one from the purchase department </li></ul><ul><li>The chartered accountant must be in the committee </li></ul>
16. 16. Question No 5 <ul><li>Twenty books are placed at random in a shelf. Find the probability that a particular pair of books shall be </li></ul><ul><li>1) always together </li></ul><ul><li>2) Never Together </li></ul>
17. 17. Question no 6 <ul><li>The letters of the word ‘failure’ are arranged at random . Find the probability that the consonants may occupy only odd positions </li></ul>