Question

DJ Cynthia is making a playlist for a radio show; she is trying to decide what 10 songs to play and in what order they should be played. If she has her choices narrowed down to 8 blues, 4 rock, 4 jazz, and 7 pop songs, and she wants to play no more than 4 pop songs, how many different playlists are possible?

Answer 1

3,770,461,100,000 different playlists are possible.

Explanation:

To solve this question, the arrangements formula and the combinations formula are used. The arrangements formula is used considering the order of the musics being played, and the combinations formula is used to find the number of possible combinations of musics.

Arrangements of n elements:

The number of possible arrangements of n elements is given by:

`A_(n) = n!`

Combinations formula:

`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

0 pop songs

10 non-pop from a set of 8 + 4 + 4 = 16.

Arrangements of 10(order in which the musics are played). So

`P(0) = 10!C_(16,10) = 10!(16!)/(6!10!) = (16!)/(6!) = 29059430400`

1 pop song:

9 non-pop from a set of 16, 1 pop from a set of 7. So

`P(1) = 10!C_(16,9)C_(7,1) = 10!(16!)/(7!9!)(7!)/(1!6!) = 290594304000`

2 pop songs:

8 non-pop from a set of 16, 2 pop from a set of 7. So

`P(2) = 10!C_(16,8)C_(7,2) = 10!(16!)/(8!8!)(7!)/(2!5!) = 980755776000`

3 pop songs:

7 non-pop from a set of 16, 3 pop from a set of 7. So

`P(3) = 10!C_(16,7)C_(7,3) = 10!(16!)/(7!9!)(7!)/(3!4!) = 1452971500000`

4 pop songs:

6 non-pop from a set of 16, 4 pop from a set of 7. So

`P(4) = 10!C_(16,6)C_(7,4) = 10!(16!)/(6!10!)(7!)/(3!4!) = 1017080100000`

How many different playlists are possible?

`T = P(0) + P(1) + P(2) + P(3) + P(4) = 29059430400 + 290594304000 + 980755776000 + 1452971500000 + 1017080100000 = 3770461100000`

3,770,461,100,000 different playlists are possible.