frans collection of twelve albums includes one rock album. fran will choose three of her albums to play on a road trip. (assume order is not important.)
a) how many different sets of three albums could she choose?
b) how many of these sets would not include the rock album?
c) how many of them would include the rock album?
a) fran can choose from $ \square $ different sets of albums.
b) there are $ \square $ sets that would not include the rock album.
c) there are $ \square $ sets that would include the rock album.

Question

Accepted Answer

# Explanation:
## Step1: Calculate the number of ways to choose 3 albums out of 12
We use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 12$ and $r=3$.
$$
\begin{align*}
C(12,3)&=\frac{12!}{3!(12 - 3)!}\
&=\frac{12!}{3!9!}\
&=\frac{12\times11\times10\times9!}{3\times2\times1\times9!}\
&=\frac{12\times11\times10}{6}\
& = 220
\end{align*}
$$
## Step2: Calculate the number of ways to choose 3 albums out of 11 (excluding the rock album)
Here $n = 11$ and $r = 3$.
$$
\begin{align*}
C(11,3)&=\frac{11!}{3!(11- 3)!}\
&=\frac{11!}{3!8!}\
&=\frac{11\times10\times9\times8!}{3\times2\times1\times8!}\
&=\frac{11\times10\times9}{6}\
&=165
\end{align*}
$$
## Step3: Calculate the number of ways that include the rock album
We know that the number of sets that include the rock album is the total number of sets of 3 albums minus the number of sets that do not include the rock album. So $220-165 = 55$

# Answer:
a) $220$
b) $165$
c) $55$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Calculate the number of ways to choose 3 albums out of 12

Step2: Calculate the number of ways to choose 3 albums out of 11 (excluding the rock album)

Step3: Calculate the number of ways that include the rock album

Answer: