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Erica and her stepmother wanted to start volunteering together. They found $7$ opportunities online, $5$ of which involved working with the elderly. $\newline$ If they randomly chose to apply to $4$ of the opportunities, what is the probability that all of them involve working with the elderly? $\newline$ 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. Erica and her stepmother wanted to start volunteering together. They found

7

opportunities online,

5

of which involved working with the elderly.

\newline

If they randomly chose to apply to

4

of the opportunities, what is the probability that all of them involve working with the elderly?

\newline

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

\newline

____

\newline

Total Outcomes Calculation: Total number of opportunities: $7$ $\newline$ Opportunities to choose: $4$ $\newline$ Calculate total possible outcomes using combinations. $\newline$ Total outcomes: $\binom{7}{4}$
Calculate Value of $7C4$ : Find the value of $_7C_4$ .
$_7C_4 = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1} = 35$
Favorable Outcomes Calculation: Number of opportunities working with the elderly: $5$ $\newline$ Number of chosen opportunities with the elderly: $4$ $\newline$ Calculate favorable outcomes using combinations. $\newline$ Favorable outcomes: $\binom{5}{4}$
Calculate Value of $5C4$ : Find the value of $_5C_4$ . $\newline$ $_5C_4$ = $\frac{5!}{4!(5-4)!}$ $\newline$ = $\frac{5!}{4!1!}$ $\newline$ = $\frac{5 \times 4!}{4! \times 1}$ $\newline$ = $5$
Calculate Probability: Calculate the probability that all chosen opportunities involve working with the elderly. $\newline$ Probability = Favorable outcomes / Total possible outcomes $\newline$ = $\frac{5}{35}$ $\newline$ = $0.1429$ when rounded to four decimal places.

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