Answer:
Different playlists possible = 18287141644800
Explanation:
Given - DJ Titus is making a playlist for a radio show; he is trying to
decide what 10 songs to play and in what order they should be
played.
Step 1 of 2 : If he has his choices narrowed down to 7 blues,
7 disco, 5 pop, and 7 reggae songs.
To find - He wants to play no more than 4 reggae songs.
How many different playlists are possible ?
Proof -
Given that he wants to play no more that 4 reggae songs.
So the possibility of choice of reggae song is 0, 1, 2, 3, 4
Now,
Case I -
If 0 reggae song is selected
⇒All 10 songs selected from 7 blue, 7 disco, 5 pop,
Number of ways = ¹⁹C₁₀ ₓ ⁷C₀ = 92,378
Case II -
If 1 reggae song is selected
⇒All 9 songs selected from 7 blue, 7 disco, 5 pop,
Number of ways = ¹⁹C₉ ₓ ⁷C₁ = 646,646
Case III -
If 2 reggae song is selected
⇒All 8 songs selected from 7 blue, 7 disco, 5 pop,
Number of ways = ¹⁹C₈ ₓ ⁷C₂ = 1,587,222
Case IV -
If 3 reggae song is selected
⇒All 7 songs selected from 7 blue, 7 disco, 5 pop,
Number of ways = ¹⁹C₇ ₓ ⁷C₃ = 1,763,580
Case V -
If 4 reggae song is selected
⇒All 6 songs selected from 7 blue, 7 disco, 5 pop,
Number of ways = ¹⁹C₆ ₓ ⁷C₄ = 949,620
So,
Total possible ways = 92,378+ 646,646+ 1,587,222+ 1,763,580+ 949,620
= 5,039,446
⇒Total possible ways = 5,039,446
Now,
Also the 10 songs selected can arranged themselves in 10! ways. ( because order of song played does not matter )
∴ we get
Different playlists possible = 10! × 5,039,446
= 18287141644800
⇒Different playlists possible = 18287141644800