Question

Howard is making a playlist for a friend; he is trying to decide what 9 songs to play and in what order they should be played.

If he has his choices narrowed down to 7 pop, 3 hip-hop, 6 country, and 7 blues songs, and he wants to play no more than 3 country songs, how many different playlists are possible? Express your answer in scientific notation rounding to the hundredths place.
__
________ x 10

Answer 1

To determine the number of different playlists Howard can make, we calculate combinations for different scenarios of country song inclusion and then multiply by the permutations of song order. The result is expressed in scientific notation.

Step-by-step explanation:

Howard's decisions for creating his playlist involves combinatorics, a field of mathematics concerned with counting and arranging objects. To calculate the total number of different playlists Howard can create, we need to consider the constraint that a maximum of three country songs can be played.

First, we choose the songs. Howard can pick from 7 pop, 3 hip-hop, 6 country (but he only wants 3 at most), and 7 blues. He needs to choose 9 songs in total. We calculate for each case distinctly: 0, 1, 2, or 3 country songs.

For example, if Howard picks 3 country songs, he then needs to choose 6 more songs from the remaining 17 non-country songs (7 pop + 3 hip-hop + 7 blues). The number of ways to choose 6 songs from 17 is given by the combination formula: '17 choose 6'. He must also consider the different orders he can arrange those 9 songs, which would be '9 factorial' possible arrangements.

We perform a similar calculation when he chooses 0, 1, or 2 country songs and then add all these possibilities together for the final count. To express our final figure in scientific notation, we must find the number that is between 1 and 10, and multiply it by 10 raised to the appropriate power.