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 n things taken r at a time will 5 things (i) always occur, (ii) never occur ?

In how many of the permutations of 10 things taken 4 at a time will (i) on thing always occur,(ii) never occur ?

The number of permutations of n things taken r at a time if 3 particular things always occur is

Find the number of permutations of n different things taken r at a time such that two specific things occur together?

STATEMENT-1 :The total number of combinations of n thing by taking some or all is 2^(n) . STATEMENT -2 : 5 identical balls can be distributed among 10 identical boxes in only one way if not more than one ball can go into a box. STATEMENT-3 : In the permutations of n things taken r at a time , the number of permutations in which m particular things occur together is ""^(n - m)P_(r - m) * ""^(r)P_(m)

The number of permutations of n different things taking r at a time when 3 particular things are to be included is a. \ ^(n-3)P_(r-3) b. \ ^(n-3)P_r c. \ ^n P_(r-3) d. \ r !^(n-3)C_(r-3)

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)

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

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

Find the total number of permutations of n different things taken not more than r at a time, when each thing may be repeated any number of times.