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

(d) $r !^{n-3} C_{r-3}$

Solution:

(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.

$\therefore$ Total number of permutations $=r !{ }^{n-3} C_{r-3}$

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