## Permutations and Combinations

### Important Formulas :

1.The number of permutations of   dissimilar objects taken   at a time is denoted by   or . 

2.The number of permutations of   dissimilar objects taken   at a time is denoted by    

3.The number of permutations of   dissimilar objects taken   at a time is denoted by   

4.The number of permutations of   dissimilar objects taken   at a time is denoted by  

5.The number of permutations of   dissimilar objects taken   at a time is denoted by  

6.Permutations when repetitions are allowed  Let   and  be positive integers such that  .Then the number of permutations of    dissimilar things taken    at a time, when repetition of things is allowed any number of time, is  .

7.Permutations when repetitions are allowed The number of permutations of    dissimilar things taken  at a time with at least one repetition  is 

8.Circular permutations : The number of circular permutations of  dissimilar things taken all at a time is ( n-1 )!

9.Circular permutations : In case of hanging type circular permutations like garlands of flowers, chains of beads etc., the number of circular permutations of   things is .

10.Permutations of objects in which some are like and the rest are different :  The number of linear permutations of   things in which  P things are similar and the rest are dissimilar is 

11.Permutations of objects in which some are like and the rest are different :  The number of linear permutations of   things in which   there are   similar things of one kind,  similar things of second kind and the rest are dissimilar is 

12.Combinations The number  of combinations of   dissimilar things  taken  at a time is denoted by 

13.Combinations  

14.Combinations 

15.Combinations For 

16.Combinations For 

17.Combinations If .

18.Combinations If .

19.If    similar things are of one kind ,  similar things  are of second kind and  similar things are of third kind, then the number of ways of selecting any number of things   ( one or more) out of them is 

20.The number of ways  of dividing  dissimilar things  into  two groups containing   and   thing is 

21.The number of ways  of dividing , dissimilar things ( are ditinct) into three groups  of   (   things is 

22.The number ways of dividing    dissimilar things into    equal groups is .

23.The number ways of  distributing     dissimilar things  equally to     persons is  .

24.The number of ways that    different things of one type and  different things of a second type where  can be arranged in a row so that no two things of the second type come together is .

25.If    are distinct primes and    are non-negative integers, then the number of positive integral divisors of   (this includes 1 and n.)

26.If    are distinct primes and    are positive integers, then the sum of distinct positive integral divisors of    is .

27.Highest power of a prime   in n! is   where  denotes the integral part.

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