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Debbie and some of her friends are going rock climbing this weekend. In preparation, Debbie purchased $8$ sports drinks to bring, including $6$ lime flavored drinks. If Debbie randomly chooses to place $4$ drinks in the red cooler, what is the probability that all of them are lime flavored? 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. Debbie and some of her friends are going rock climbing this weekend. In preparation, Debbie purchased

8

sports drinks to bring, including

6

lime flavored drinks. If Debbie randomly chooses to place

4

drinks in the red cooler, what is the probability that all of them are lime flavored? Write your answer as a decimal rounded to four decimal places. ____

Calculate Total Combinations: First, figure out the total number of ways to choose $4$ drinks out of $8$ . This is a combination problem, so we use the combination formula: $C(n, k) = \frac{n!}{k!(n-k)!}$ $C(8, 4) = \frac{8!}{(4!(8-4)!)} = \frac{(8\times7\times6\times5)}{(4\times3\times2\times1)} = 70$ ways.
Find Lime Drink Combinations: Next, find the number of ways to choose $4$ lime drinks out of the $6$ available. $\newline$ $C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{(6\times5)}{(2\times1)} = 15$ ways.
Calculate Probability: Now, calculate the probability that all $4$ drinks are lime flavored. $\newline$ Probability = Number of favorable outcomes / Total number of outcomes $\newline$ Probability = $\frac{15}{70}$
Convert to Decimal: Finally, convert the fraction to a decimal and round to four decimal places. $\newline$ Probability = $\frac{15}{70} \approx 0.2143$

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