Question

this quiz will test your knowledge on permutations and combinations. value: 1 there are seventeen songs on your playlist. nine of them are country and eight are pop. with random shuffle and no repetition, you listen to nine songs. what is the probability that you listened to exactly five country songs? a. 36% b. 32% c. 29% d. 35%

Explanation:

Step1: Calculate total number of ways to choose 9 songs out of 17

The number of combinations of choosing $r$ items from $n$ items is given by the formula $C(n,r)=\frac{n!}{r!(n - r)!}$. Here, $n = 17$ and $r=9$, so $C(17,9)=\frac{17!}{9!(17 - 9)!}=\frac{17!}{9!8!}=\frac{17\times16\times15\times14\times13\times12\times11\times10}{8\times7\times6\times5\times4\times3\times2\times1}=24310$.

Step2: Calculate number of ways to choose 5 country - songs out of 9 and 4 pop - songs out of 8

The number of ways to choose 5 country - songs out of 9 is $C(9,5)=\frac{9!}{5!(9 - 5)!}=\frac{9!}{5!4!}=\frac{9\times8\times7\times6}{4\times3\times2\times1}=126$.
The number of ways to choose 4 pop - songs out of 8 is $C(8,4)=\frac{8!}{4!(8 - 4)!}=\frac{8!}{4!4!}=\frac{8\times7\times6\times5}{4\times3\times2\times1}=70$.
By the multiplication principle, the number of ways to choose 5 country - songs and 4 pop - songs is $C(9,5)\times C(8,4)=126\times70 = 8820$.

Step3: Calculate the probability

The probability $P$ is the number of favorable outcomes divided by the number of total outcomes. So $P=\frac{8820}{24310}\approx0.36$.

Answer:

a. 36%