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Annie and her best friend wanted to paint their apartment. In their storage closet, they found $8$ cans of paint. Annie remembered that $6$ of the cans contained beige paint, but none of the cans had labels. $\newline$ If Annie randomly opened $5$ cans, what is the probability that all of them contain beige paint? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ ____ $\newline$

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Find probabilities using combinations and permutations

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Q. Annie and her best friend wanted to paint their apartment. In their storage closet, they found

8

cans of paint. Annie remembered that

6

of the cans contained beige paint, but none of the cans had labels.

\newline

If Annie randomly opened

5

cans, what is the probability that all of them contain beige paint?

\newline

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

\newline

____

\newline

Total Cans Calculation: Total number of cans: $8$ $\newline$ Cans to be opened: $5$ $\newline$ Calculate the total number of ways to choose $5$ cans from $8$ . $\newline$ Total outcomes: $_{8}C_{5}$
Calculate Total Ways: Find the value of ${}_8C_5$ . ${}_8C_5 = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} = 56$
Beige Paint Cans: Number of beige paint cans: $6$ $\newline$ Cans of beige paint to be opened: $5$ $\newline$ Calculate the number of ways to choose $5$ beige cans from $6$ . $\newline$ Favorable outcomes: $_{6}C_{5}$
Calculate Beige Cans: 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 $5$ opened cans contain beige paint. $\newline$ Probability = $\frac{\text{Favorable outcomes}}{\text{Total outcomes}}$ $\newline$ = $\frac{6}{56}$ $\newline$ $\approx 0.1071$

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