Question

A rock group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if they have 15 songs to pick from?

Answer 1

We need to find the number of combinations of 3 songs out of 15 songs.

The combination of r items out of a total of n items is given by the formula:

`C(n,r)=(n!)/(r!(n-r)!)`

where

`n!=n(n-1)(n-2)...(2)(1)`

In this problem, we have:

`\begin{gathered} n=15 \\ r=3 \end{gathered}`

Thus, we obtain:

`\begin{gathered} C(15,3)=(15!)/(3!(15-3)!) \\ \\ C(15,3)=(15\cdot14\cdot13(12!))/(3\cdot2\cdot1(12!)) \\ \\ C(15,3)=(15)/(3)\cdot(14)/(2)\cdot13 \\ \\ C(15,3)=5\cdot7\cdot13 \\ \\ C(15,3)=455 \end{gathered}`

Answer: The number of ways they can choose their set of songs is 455.