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The Danville High School Speech and Debate Club hosted its annual car wash fundraiser. Each club member brought a bottle of car wash soap, so there were $8$ total bottles. $6$ of the bottles contained green soap. $\newline$ If a club member randomly selects $5$ bottles to pour into the first soap bucket, what is the probability that all of them contain green soap? $\newline$ 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. The Danville High School Speech and Debate Club hosted its annual car wash fundraiser. Each club member brought a bottle of car wash soap, so there were

8

total bottles.

6

of the bottles contained green soap.

\newline

If a club member randomly selects

5

bottles to pour into the first soap bucket, what is the probability that all of them contain green soap?

\newline

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

Total Bottles and Selection: Total number of bottles: $8$ $\newline$ Bottles to choose for the soap bucket: $5$ $\newline$ Calculate the total number of ways to choose $5$ bottles from $8$ . $\newline$ Total outcomes: $_{8}C_{5}$
Calculate Total Ways: Find the value of ${}_8C_5$ . ${}_8C_5 = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} = 56$
Green Bottles Selection: Number of green soap bottles: $6$ $\newline$ Number of chosen green soap bottles: $5$ $\newline$ Calculate the number of ways to choose $5$ green bottles from $6$ . $\newline$ Favorable outcomes: $_{6}C_{5}$
Calculate Green Bottles Ways: Find the value of ${}_6C_5$ . ${}_6C_5 = \frac{6!}{5!(6-5)!} = \frac{6!}{5!1!} = \frac{6 \times 5!}{5! \times 1} = 6$
Calculate Probability: Calculate the probability that all selected bottles are green. $\newline$ Probability = Favorable outcomes / Total possible outcomes $\newline$ = $\frac{6}{56}$ $\newline$ $\approx 0.1071$

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