4. Four friends are tired of their keychains and decide to redistribute them so that each of them has a different one from the one they had before. In how many different ways can they exchange the keychains?
(A) 6
(B) 9
(C) 7
(D) 8
(E) 10
Problem 4. The answer is (B).

The problem is equivalent to permuting the numbers 1,2,3,4 in such a way that the number $i$ does not appear at position $i$. The nine and only possibilities to do this are:

$$
2,1,4,3 \quad 2,3,4,1 \quad 2,4,1,3 \quad 3,1,4,2 \quad 3,4,1,2 \quad 3,4,2,1 \quad 4,1,2,3 \quad 4,3,1,2 \quad 4,3,2,1
$$

(Problem proposed by A. Dal Zotto)