Please help with this permutations question. I'm trying to use the permutation formula to calculate it but don't know where to begin: $$\frac{n!}{(n-r)!}$$

Here's the problem: Jane is choosing photos to display in 2 frames. Each frame holds 4 photos. She is choosing from a number of family photos to arrange in the first frame and a number of vacation photos to arrange in the second frame. Which numbers of family photos and vacation photos would result in more than 500,000 ways to arrange the photos in the frames?

Possible answers are:

[ ] 5 family photos and 9 vacation photos
[X] 6 family photos and 8 vacation photos
[X] 7 family photos and 7 vacation photos
[ ] 10 family photos and 4 vacation photos

I've indicated the correct answers above but not sure how those are correct. Thanks.


Consider just the family photos. If there are $n$ family photos, how many ways are there to arrange those photos in the first frame? If there are $m$ vacation photos, how many ways are there to arrange those in the second frame? (To answer each of these questions, the formula in your question will be handy). Finally, how can you compute the total number of ways from the two answers above?

  • $\begingroup$ Would the formula for the family photos be this for the first option: $$\frac{5!}{(5-4)!}$$ and this for the vacation photos: $$\frac{9!}{(9-4)!}$$ $\endgroup$ – Alex Apr 17 '14 at 1:21
  • $\begingroup$ Then multiply the two results: $$(120)(3024)$$ to get 362880 permutations. $\endgroup$ – Alex Apr 17 '14 at 1:29
  • 1
    $\begingroup$ Yes, that's right. $\endgroup$ – rogerl Apr 17 '14 at 12:22

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