47. In how many ways can 10 different photographs be placed in six different envelopes, no envelope remaining empty? Hint: An easy way to do this problem is to use the following version of the inclusion-exclusion principle: Let A1, A2, . . . , An be n subsets of a finite set # with N elements. Let N (Ai) be the number of elements of Ai, 1 ≤ i ≤ n, and N (Ac i) = N − N (Ai). Let Sk be the sum of the elements of all those intersections of A1, A2, . . . , An that are formed of exactly k sets. That is, S1 = N (A1) + N (A2) + · · · + N (An), S2 = N (A1A2) + N (A1A3) + · · · + N (An−1An), and so on. Then N (Ac 1Ac 2 ··· Ac n) = N − S1 + S2 − S3 + · · · + (−1) nSn. To solve the problem, let N be the number of ways that 10 different photographs can be placed in six different envelopes, allowing for the possibility of empty envelopes. Let Ai be the set of all situations in which envelope i is empty. Then the desired quantity is N (Ac 1Ac 2 ··· Ac 6).