In how many of the permutations of `n` thing taken `r` at a time will three given things occur?

In how many of the permutations of 12 things taken 6 at a time will 3 particular things (a) always occur b) never occur ?

In how many of the permutations of 12 things taken 3 at time will one particular thing (a) always occur (b) never occur ?

Find the number of permutations of 20 things taken 5 at a time so that one particular thing will always occur

In how many of the permutations of the letters in the word SYNDICATE taken 4 at a time , there shall be at least one vowels?

In how many of the permutations of the letters in the word SYNDICATE taken 4 at a time , there shall be no vowel

Find the number of permutations of 20 things taken 5 at a time so that one particular thing will never occur.

The number of permutations of n different things taken r at a time in which 4 particular things never occur is -

If a denotes the number of permutations of (x+2) things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of x-11 things taken all at a time such that a=182b c , find the value of xdot

Let .^(n)P_(r) denote the number of permutations of n different things taken r at a time . Then , prove that 1+1.^(1)P_(1)+2.^(2)P_(2)+3.^(3)P_(3)+...+n.^(n)P_(n)=.^(n+1)P_(n+1) .

Find the number of permutations of n different things taken r at time so that m(ltr) particular things will always occur