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There are $9$ college students running for student government class president. The candidates include $7$ education majors. $\newline$ If $6$ of the candidates are randomly chosen to give the first $6$ speeches, what is the probability that all of them are education majors? $\newline$ Write your answer as a decimal rounded to four decimal places.

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Find probabilities using combinations and permutations

Full solution

Q. There are

9

college students running for student government class president. The candidates include

7

education majors.

\newline

If

6

of the candidates are randomly chosen to give the first

6

speeches, what is the probability that all of them are education majors?

\newline

Write your answer as a decimal rounded to four decimal places.

Calculate Total Outcomes: Determine the total number of ways to choose $6$ candidates out of $9$ . $\newline$ We use the combination formula, which is ${}_{n}C_{r} = \frac{n!}{r!(n-r)!}$ , where $n$ is the total number of items, $r$ is the number of items to choose, and ＂!＂ denotes factorial. $\newline$ Substitute $9$ for $n$ and $6$ for $r$ to find the total number of ways to choose $6$ candidates out of $9$ . $\newline$ Total outcomes: ${}_{9}C_{6} = \frac{9!}{6!(9-6)!}$
Calculate Value of $9C6$ : Calculate the value of $_9C_6$ .
$_9C_6 = \frac{9!}{6!3!}$
= $\frac{9 \times 8 \times 7 \times 6!}{6! \times 3 \times 2 \times 1}$
= $\frac{9 \times 8 \times 7}{3 \times 2 \times 1}$
= $\frac{504}{6}$
= $84$
Total possible outcomes = $84$
Determine Favorable Outcomes: Determine the number of ways to choose $6$ education majors out of $7$ . $\newline$ Again, we use the combination formula. $\newline$ Substitute $7$ for $n$ and $6$ for $r$ to find the number of ways to choose $6$ education majors out of $7$ . $\newline$ Favorable outcomes: $\binom{7}{6} = \frac{7!}{(6!(7-6)!)}$
Calculate Value of $7C6$ : Calculate the value of $_7C_6$ . $\newline$ $_7C_6 = \frac{7!}{(6!1!)}$ $\newline$ $= \frac{(7 \times 6!)}{(6! \times 1)}$ $\newline$ $= 7$ $\newline$ Favorable outcomes = $7$
Calculate Probability: Calculate the probability that all $6$ chosen candidates are education majors. $\newline$ Probability = Favorable outcomes / Total possible outcomes $\newline$ = $\frac{7}{84}$ $\newline$ = $\frac{1}{12}$
Convert to Decimal: Convert the probability to a decimal rounded to four decimal places. $\newline$ Probability $\approx 0.0833$

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