If the number of arrangements of `(n-1)` things taken from `n` different things is `k` times the number of arrangements of `(n - 1)` things taken from n things in which two things are identical then the value of `k` is

Find the number of permutations of n things taken r at a time in which two particular things : always occur.

Find the number of permutations of n things taken r at a time in which two particular things : never occur.

Find the number of permutations of n things, taken r at a time, in which 3 particular things : always occur.

If six times the number of permutations of n things taken 3 at a time is equal to seven times the number of permutations of (n-1) things chosen 3 at a time, find n.

Find the number of permutations of n things, taken r at a time, in which 3 particular things : will never occur.

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 things taken r at a time in which two particular things : (i) always occur (ii) never occur.

If a denotes the number of permutations of x + 2 things taken all at a time, b denotes the number of permutations of x things taken 1 1 at a time and c denotes the number of permutations of x - 1 1 things taken all at a time such that a = 182 be, then the value of x is

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