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

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

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

2.The number of permutations of 8 things taken at a time is 1680, Then r=

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

If the number of permutations of n different things taken 4 at a time in which one particular thing does not occur is equal to that in which it does occur,find n .