I have 30 photos to post on my website. I\'m planning to post these on two web p

Question

I have 30 photos to post on my website. I'm planning to post these on two web pages, one marked "Friends" and the other marked "Family". No photo may go on both pages, but every photo will end up on one or the other. Conceivably, one of the pages may be empty. Please answer these two questions:

a. In how many ways can I post these photos to the web pages if the order in which the photos appear on those pages matters?

b. In how many ways can I post these photos to the web pages if the order in which the photos appear on those pages does not matter?

Explanation / Answer

a)

Let’s consider a simpler example. Instead of 30 photos, we only have 3. First let’s consider all 3 photos posted

on the “Friends” page. We would have 3! possible arrangements. We have another 3! possible arrangements if

we post them all to the “Family” page. If we post 2 on “Friends”, and 1 on “Family”, we have another

3! possibilities, and another when we post 2 on “Family” and 1 on “Friends”. So altogether we have 4·3! = 4!

possible arrangements.

Now consider 4 photos. There would be 5·4! = 5 possible arrangements. Extrapolating to 30 photos, we

would get 31·30! = 31! possible arrangements.

b) In the case where we put all the picture on one page, and not the other, since order doesn’t matter, each of

those counts as 1 arrangement. We can rephrase the problem as the number of possible ways to choose a

page for each picture. In the 3 photo situation, it comes out to 23 = 8. In the 30 photo situation, it comes

out to 230 ways.

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