Question

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) ⁿP_{r − 3}

(d) r ! ^{n − 3}C_{r − 3}

Answer 1

(d) r ! ^{n − 3}C_{r − 3}
Here, we have to permute n things of which 3 things are to be included.
So, only the remaining (n$-$3) things are left for permutation, taking (r$-$3) things at a time. This is because 3 things have already been included.
But, these r things can be arranged in r! ways.
∴ Total number of permutations = r ! ^{n − 3}C_{r − 3}