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Fri 21 Apr, 2006 10:19 pm
Jeffrey is a DJ at a local radio station. For the second hour of his shift, he must choose all his music from the top 100 songs for thw eek. Jeffrey will play exactly 12 songs during this hour.
a) How many different stacks of discs could Jeffrey pull from the station's collection if he chooses at least 10 songs that are in positions 15 to 40 on the charts?
b)Jeffrey wants to start his second hour with a hard-rock song and finish with a pop classic. How many different play lists can Jeffrey prepare if he has chosen 4 hard rock songs, 5 soul pieces, and 3 pop classics?
c)Jeffrey has 8 favourite songs currently on the top 100 list. How many different subsets of these songs could he choose to play during his shift?
thanks in advance
(a) (# ways to choose 10 from 15-40) * (# ways to choose 2 from what's left) * (# ways to arrange the 12 songs if order matters)
(b) (# ways to choose a hard-rock song) * (# ways to choose a pop classic) * (# ways to arrange the 10 middle songs)
(c) Since 8<12, he can choose all possible subsets of the 8 songs. Hint: a song is either in a subset, or it is not.
For (a), there are 26 songs between 15 and 40 inclusive. For his 10 songs, there are 26!/16! permutations for those slots. He now has 90 songs remaining for the remaining 2 slots. There are 90!/88! permuations (or just 90*89) for those two positions. Since you want combinations, not permutations, you need to divide the product by 12!. So you are left with (26!/16!)*(90*89)/(12!)
If you understand that, post your best effort for (b) and (c). If not, post back what you don't understand.
engineer:
(26!/16!)*(90*89)/(12!) is not the answer.
It equates to 322,325,737.5.
The ten (and two) songs were not permuted across all 12 positions; so dividing by 12! doesn't make sense.
10! * 2! != 12!
Multiply your answer by 12! / (10! * 2!) to get my (combinations) answer.
10 songs from 15-40: C(26,10) = 26! / (16! * 10!) = 5,311,735
2 songs from 90: C(90,2) = 90! / (88! * 2!) = 4,005
If order doesn't matter (combinations), the answer is:
5,311,735 * 4,005 = 21,273,498,675
If order does matter (permutations), the answer is:
21,273,498,675 * 12! = 10,190,039,902,922,880,000
Thanks for the correction Mark.
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