there are 9 students performing in the talent show, 7 of whom will play an instrument. if 6 students are…

there are 9 students performing in the talent show, 7 of whom will play an instrument. if 6 students are randomly chosen to perform during the first section of the show, what is the probability that all of them will play an instrument? write your answer as a decimal rounded to four decimal places.

there are 9 students performing in the talent show, 7 of whom will play an instrument. if 6 students are randomly chosen to perform during the first section of the show, what is the probability that all of them will play an instrument? write your answer as a decimal rounded to four decimal places.

Answer

Explanation:

Step1: Calculate total combinations

We use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 9$ (total students) and $r=6$ (students chosen). So $C(9,6)=\frac{9!}{6!(9 - 6)!}=\frac{9!}{6!3!}=\frac{9\times8\times7}{3\times2\times1}=84$.

Step2: Calculate favorable combinations

We want all 6 - student group to be from the 7 instrument - playing students. Using the combination formula with $n = 7$ (instrument - playing students) and $r = 6$ (students chosen), we get $C(7,6)=\frac{7!}{6!(7 - 6)!}=\frac{7!}{6!1!}=7$.

Step3: Calculate probability

The probability $P$ is the number of favorable combinations divided by the total number of combinations. So $P=\frac{C(7,6)}{C(9,6)}=\frac{7}{84}=\frac{1}{12}\approx0.0833$.

Answer:

$0.0833$