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Discrete math lists: MP3 player playlist, photos, rooks

I want to create two play lists on my MP3 player from my collection of 500 songs. One playlist is titled "Exercise" for listening in the gym and the other is titled "Relaxing" for leisure time at home. I want 20 different songs on each of these lists.

In how many ways can I load songs onto my MP3 player if I allow a song to be on both playlists?
And, how many ways can I load the songs if I want the two lists to have no overlap?

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.

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?

In how many ways can a black rook and a white rook be placed on different squares of a chess oard such that neither is attacking the other? (In other words, they cannot be in the same row or the same column of the chess board. A standard chess board is 8x8).

Solution Preview

Please see the attached file.

Say we want to select a subset of m elements out of a set N, when the order of elements is important (that is, the combination {1,2,3} is different than {2,3,1}) and there no element can be selected more than once (no replacements).
We denote the number of possible permutations as
For the first place is the subset we have N possible selections. For the second place of the subset we have N-1 possible selections. For the third place we have N-2 possible selections and it is easy to see that for the kth place we have N-k+1 possible selections.
Therefore, the number of possible different subsets the product of all the possible ways we can fill a certain position in the subset:
(1.1)
We can write it as:
(1.2)
Or more concise, the number of permutations of selecting n elements out of a list of N elements is:
(1.3)
Example: We have a set {1,2,3,4} of N=4 elements and we need to select 2 elements to form a subset. The number of possible permutations is:
(1.4)
And they are:
{1,2},{1,3},{1,4},{2,1},{2,3},{2,4},{3,1},{3,2},{3,4},{4,1},{4,2},{4,3}
Now, assume we don't care about the order of the elements ...

Solution Summary

The discrete math lists for MP# player playlists, photos and rooks are examined.

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