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

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)

Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

In how many of the permutations of n things taken r at a time will 5 things (i) always occur, (ii) never occur ?

Prove that the no. of permutations of n different objects taken r at a time in which two specified objects always occur together 2!(r-1) (n-2)P_(r-2)

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 thing taken r at a time will three given things occur?