DJ Ella is making a playlist for an internet radio show; she is trying to decide what 10 songs to play and in what order they should be played. Step 1 of 2 : If she has her choices narrowed down to 7 blues, 6 disco, 8 pop, and 4 hip-hop songs, and she wants to play no more than 4 disco songs, how many different playlists are possible

She can make up her 10 song playlist in 3195116 ways

Step-by-step explanation:

7 blues, 6 disco, 8 pop, 4 hip hops

Total songs = 25

She wants to play no more than 4 disco songs in a playlist of 10 songs

Meaning the music can be grouped as disco and other music genres.

Disco songs = 6

Other genres = 19

She can make up her 10 song playlist in any of the following ways.

- 10 other genres, 0 disco song

This playlist can be made in

¹⁹C₁₀ ways = 92378 ways

- 9 other genres, 1 disco song

This playlist can be made in

¹⁹C₉ * ⁶C₁ = 92378 * 6 = 554268 ways

- 8 other genres, 2 disco song

This playlist can be made in

¹⁹C₈ * ⁶C₂ = 75582 * 15 = 1133730 ways

- 7 other genres, 3 disco song

This playlist can be made in

¹⁹C₇ * ⁶C₃ = 50388 * 20 = 1007760

- 6 other genres, 4 disco song

This playlist can be made in

¹⁹C₆ * ⁶C₄ = 27132 * 15 = 406980

Total number of ways = 92378 + 554268 + 1133730 + 1007760 + 406980 = 3195116 ways