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Over the past year, the chef at a popular restaurant has featured $8$ different entrées, $6$ of which are soups. If the chef creates a new menu by randomly selecting $4$ previously featured entrées, what is the probability that all of them are soups? $\newline$ Write your answer as a decimal rounded to four decimal places._________.

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Find probabilities using combinations and permutations

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Q. Over the past year, the chef at a popular restaurant has featured

8

different entrées,

6

of which are soups. If the chef creates a new menu by randomly selecting

4

previously featured entrées, what is the probability that all of them are soups?

\newline

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

Calculate Total Entrees Combinations: Calculate the total number of ways to choose $4$ entrées from $8$ . Use the combination formula: $C(n, k) = \frac{n!}{k!(n-k)!}$ $C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{(8\times7\times6\times5)}{(4\times3\times2\times1)} = 70$ .
Calculate Soups Combinations: Calculate the number of ways to choose $4$ soups from the $6$ available. $\newline$ Use the combination formula again: $C(n, k) = \frac{n!}{k!(n-k)!}$ $\newline$ $C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{(6\times5)}{(2\times1)} = 15$ .
Find Probability: Find the probability by dividing the number of ways to choose $4$ soups by the total number of ways to choose $4$ entrées. $\newline$ Probability = Number of ways to choose $4$ soups / Total number of ways to choose $4$ entrées $\newline$ Probability = $15 / 70$ .
Convert to Decimal: Convert the fraction to a decimal and round to four decimal places. $\newline$ Probability = $\frac{15}{70} \approx 0.2143$ .

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