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A band has a portfolio of 15 songs. 5 are acoustic. For a concert the band will play 10distint songs, 4 will be acoustic. The 4 acoustic will be played consecutively during the 2nd half, but withou ending in anacoustic song. How many different sequences of the songs are possible?

 Dec 18, 2018

\(\text{looking at the 2nd half of the show first}\\ \text{the songs will be AAAA!A}\\ \text{there are }4!=24 \text{ ways to arrange the 4 acoustic songs, and}\\ \text{10 ways to select the !A song at the end}\\ \text{this is then 240 ways to arrange the 2nd half of the show}\\ \text{the first half of the show is then 5 non-acoustic songs chosen from 9}\\ \text{since we care about order there are }\dbinom{9}{5}5! = 15120\\ \text{ arrangements for the first half} \\ \text{so in total we have}\\ 240 \times 15120 = 3628800 \text{ possible concert arrangements}\)

 Dec 18, 2018

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