Prove that the no. of mutually distinguishable permutations of n things ; taken all at a time of which p are alike of one kind; q alike of second kind such that `p+q=n` is `(n!)/(p!.q!)`

The no.of all permutation of n distinct things taken all at a time is n!

Prove that the number of combinations of n things taken r at a time in which p particular things always occur is .^(n-p)C_(r-p)

The number of linear permutations of n' things in which there are Plike thing of one kind,g like things of second kinds 'like things of third kind and the rest are different is (n!)/(p!q!r!) The number of ways of arranging the letter of the word ARRANGE so,that.The two R' come together is

The total number of ways in which a selection(one or more) can be made of p+q+r things of which p are all alike, q all alike, r all alike is ( p + 1 ) ( q + 1 ) ( r + 1 )-1 II: The number of permutations of 'n' things taken together when 'p' of the things are alike of one kind, 'q' of them alike of a second kind, 'r' of them alike of a third kind and the rest all different is (n!)/(p!q!r!3!) Which of the above statement is true?

The number of ways of selecting ' n ' things out of '3n ' things of which 'n ' are of one kind and alike and 'n ' are of second kind and alike and the rest unlike is :

If P(n, n) denotes the number of permutations of n different things taken all at a time then P(n, n) is also identical to , where 0 le r le n

In the permutations of n things r, taken together, the number of permutations in which m particular things occur together is ""^(n-m)P_(r-m)xx""^(r)P_(m) .

Let P_(n) denote the number of permutation of n distinct things taken all at a time and x_(n)=^(n+5)C_(4)-((143)/(96))((P_(n+5))/(P_(n+3))) (where n in N .The possible value of n for which x_(n) is negative,can be