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Dana is shopping for a costume at a thrift store. There are $8$ costumes hanging on the rack, including $6$ food-shaped costumes. If the costumes all look like the right size, and Dana randomly selects $5$ to try on, what is the probability that all of them are food-shaped costumes? 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. Dana is shopping for a costume at a thrift store. There are

8

costumes hanging on the rack, including

6

food-shaped costumes. If the costumes all look like the right size, and Dana randomly selects

5

to try on, what is the probability that all of them are food-shaped costumes? Write your answer as a decimal rounded to four decimal places.._________

Costumes Total and Selection: Total number of costumes: $8$ Costumes Dana wants to try on: $5$ Calculate the total number of ways to choose $5$ costumes from $8$ . Use the combination formula: ${}_8C_5$ .
Calculate Total Combinations: Find the value of $_8C_5$ . $_8C_5 = \frac{8!}{5!(8-5)!} =\frac{8!}{5!3!} =\frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} = 56$
Food-Shaped Costumes Selection: Number of food-shaped costumes: $6$ Number of food-shaped costumes Dana wants to try on: $5$ Calculate the number of ways to choose $5$ food-shaped costumes from $6$ . Use the combination formula: $_6C_5$ .
Calculate Food-Shaped Combinations: 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 $5$ costumes are food-shaped. $\newline$ Probability = Favorable outcomes / Total possible outcomes $\newline$ = $\frac{6}{56}$ $\newline$ $\approx 0.1071$

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