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Jim is shopping for a costume at a thrift store. There are $8$ costumes hanging on the rack, including $6$ superhero costumes. $\newline$ If the costumes all look like the right size, and Jim randomly selects $5$ to try on, what is the probability that all of them are superhero costumes? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ ____ $\newline$

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Find probabilities using combinations and permutations

Full solution

Q. Jim is shopping for a costume at a thrift store. There are

8

costumes hanging on the rack, including

6

superhero costumes.

\newline

If the costumes all look like the right size, and Jim randomly selects

5

to try on, what is the probability that all of them are superhero costumes?

\newline

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

\newline

____

\newline

Calculate probability of first costume: First, calculate the probability of picking $1$ superhero costume from the rack. $\newline$ There are $6$ superhero costumes out of $8$ total costumes. $\newline$ So, the probability for the first costume is $\frac{6}{8}$ .
Calculate probability of second costume: Now, since one superhero costume is taken, there are $5$ superhero costumes left and $7$ costumes in total. $\newline$ The probability for the second costume is $\frac{5}{7}$ .
Calculate probability of third costume: For the third costume, there are now $4$ superhero costumes and $6$ total costumes left. $\newline$ The probability for the third costume is $\frac{4}{6}$ .
Calculate probability of fourth costume: Moving on to the fourth costume, there are $3$ superhero costumes and $5$ total costumes left. $\newline$ The probability for the fourth costume is $\frac{3}{5}$ .
Calculate probability of fifth costume: Finally, for the fifth costume, there are $2$ superhero costumes and $4$ total costumes left. $\newline$ The probability for the fifth costume is $\frac{2}{4}$ .
Find overall probability: To find the overall probability of all $5$ costumes being superhero costumes, multiply the individual probabilities together. $\newline$ $(\frac{6}{8}) \times (\frac{5}{7}) \times (\frac{4}{6}) \times (\frac{3}{5}) \times (\frac{2}{4}) = \frac{6\times5\times4\times3\times2}{8\times7\times6\times5\times4} = \frac{720}{6720}$

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