Originally Posted by

**riemann** There are 10 persons and 10 prizes, each prize belonging to a distinct person. *Find the number of ways of distributing prizes among all the persons such that 3 particular persons never get their respective prizes in any of the distributions*. Each person must get 1 prize.

For example:

Suppose the persons are named as A,B,C,D,E,F,G,H,I,J and their respective prizes are numbered 1,2,3,4,5,6,7,8,9,10. (Its like prize 1 goes to person A, prize 2 goes to person B and so on.) Now,

A-1 B-2 C-3 D-6 E-7 F-4 G-5 H-8 I-9 J-10 (Here D, E, F are the 3 particular persons who didn't get their respective prizes. The rest may or may not get their respective prizes) is one such possible distribution.

In other words, if it were 4 people, 4 prizes, and 2 "particular" persons never get the correct prize, here's what I would say:

Call the people A,B,C,D... call their prizes a,b,c,d

Our two "particular" people are A and B

The possibilities:

Ab, Ba, Cc, Dd

Ab, Ba, Cd, Dc

Ab, Bc, Ca, Dd

Ab, Bc, Cd, Da

Ab, Bd, Ca, Dc

Ab, Bd, Cc, Da

Ac, Ba, Cb, Dd

Ac, Ba, Cd, Db

Ac, Bd, Ca, Db

Ac, Bd, Cb, Da

Ad, Ba, Cb, Dc

Ad, Ba, Cc, Db

Ad, Bc, Ca, Db

Ad, Bc, Cb, Da

So... 14.