Lucy's Playlist
Lucy is planning to run for an hour, and she's using her iPod for the countdown, so she has to create a 60-minute playlist.
Her iPod contains:
- six 5-minute songs;
- five 6-minute songs;
- four 8-minute songs.
Count the number of possibles playlists where the order doesn't matter.
Details and Assumptions :
- She must use up all 60 minutes for the playlist.
The answer is 621.
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1 solution
Discussions for this problem are now closed
Hi, do you have a much better method than solving to find the solutions by hand ? I wasted a lot of time by the time I solved your problem .
Thanks for the same .
@brian charlesworth @Calvin Lin @Chew-Seong Cheong @abdulrahman khaled Help needed !!!
Not really, you could convert it to a generating function question, and have to find the coefficient of x 6 0 in
( 1 + ( 1 6 ) x 5 + ( 2 6 ) x 1 0 + … + ( 6 6 ) x 3 0 ) × ( 1 + ( 1 5 ) x 6 + ( 2 5 ) x 1 2 + … + ( 5 5 ) x 3 0 ) × ( 1 + ( 1 4 ) x 8 + ( 2 4 ) x 1 6 + … + ( 4 4 ) x 3 2 )
Though, in manually calculating this, you are no better off than the original solution, since you are doing the same calculations.
Thanks sir, I had thought of using a generation function but didn't know how to create one for this particular question .
I've written a code to solve the equation , I hope somebody with more knowledge in solving such problems would help us
Can you please explain why 0 ≤ x , y , z ≤ 5 ?I mean how did you decide on the upper limit of 5 . I understood rest of your solution.
I'm sorry , it's a mistake , I'm fixing it
It's kinda rude, but is the last solution (6, 5, 0)? Sorry, that might be a little thing, but yeah, maybe it should be corrected.
Anyway, the last part is still okay. Don't worry. :D
fixed it ! thank's
But the question here is asking for possibilities where order DOES NOT matter
sure , that's why I used n choose k = n!/(k!(n-k)!) , not n!/(n-k)! in other words , divide by all the possible permutations (every possible order for the same set)
let X be the number of 5min songs , Y the number of 6 min songs and Z the number of 8min songs In order to get a 60min playlist we need to solve this equation : 5 x + 6 y + 8 z = 6 0 where 0 ≤ x ≤ 6 0 ≤ y ≤ 5 0 ≤ z ≤ 4 so we get those four solutions : ( 2 , 3 , 4 ) , ( 4 , 4 , 2 ) , ( 6 , 1 , 3 ) a n d ( 6 , 5 , 0 ) The number of possibles playlists is : ( 2 6 ) × ( 3 5 ) × ( 4 4 ) + ( 4 6 ) × ( 4 5 ) × ( 2 4 ) + ( 6 6 ) × ( 1 5 ) × ( 3 4 ) + ( 6 6 ) × ( 5 5 ) × ( 0 4 ) 1 5 0 + 4 5 0 + 2 0 + 1 = 6 2 1