Explanation:

Suppose the letters be L1, L2, L3 and L4, and the envelopes be E1, E2, E3, and E4.

First, since L1 cannot be assigned to E1, assign L1 to E2. Then list out the possibilities for the remaining three letters such that NO letter is placed in its correct envelope.

The only three possibilities are

(L2 to E1, L3 to E4, L4 to E3),

(L2 to E3, L3 to E4, L4 to E1), and

(L2 to E4, L3 to E1, L4 to E3).

So , there are 3 possible ways to assign the remaining letters when L1 assigned to E2. So, there must be 3 possible ways to assign the remaining letters with L1 assigned to E3, and there must be 3 possible ways to assign the remaining letters with L1 assigned to E4.

Therefore, there are a total of 3 + 3 + 3 = 9 ways the secretary can place the letters in the envelopes.

Hence, option 4 is the correct answer.