Question

steve wants to download a selection of new music. how many ways can steve select four rock songs, two alternative songs, and five rap songs from a list of seven rock songs, six alternative songs, and ten rap songs? there are ways that steve can select his songs.

Explanation:

Step1: Use combination formula

The combination formula is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n\) is the total number of items and \(r\) is the number of items to be chosen.
For choosing \(4\) rock songs out of \(7\): \(C(7,4)=\frac{7!}{4!(7 - 4)!}=\frac{7\times6\times5\times4!}{4!\times3!}=\frac{7\times6\times5}{3\times2\times1}=35\).

Step2: Choose alternative songs

For choosing \(2\) alternative songs out of \(6\): \(C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5\times4!}{2!\times4!}=\frac{6\times5}{2\times1}=15\).

Step3: Choose rap songs

For choosing \(5\) rap songs out of \(10\): \(C(10,5)=\frac{10!}{5!(10 - 5)!}=\frac{10\times9\times8\times7\times6\times5!}{5!\times5!}=\frac{10\times9\times8\times7\times6}{5\times4\times3\times2\times1}=252\).

Step4: Use multiplication principle

Since the choices of rock, alternative, and rap songs are independent events, the total number of ways is the product of the number of ways of each choice. So the total number of ways \(N = C(7,4)\times C(6,2)\times C(10,5)\).
Substitute the values: \(N=35\times15\times252\).
\(35\times15 = 525\), then \(525\times252=(500 + 25)\times252=500\times252+25\times252\).
\(500\times252 = 126000\), \(25\times252=6300\).
\(N=126000+6300=132300\).

Answer:

\(132300\)