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Peter performs $9$ exercises over the span of a week. $7$ of his exercises involve biking. $\newline$ If he randomly chose $6$ exercises to do on Monday, what is the probability that all of them involve biking? $\newline$ 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. Peter performs

9

exercises over the span of a week.

7

of his exercises involve biking.

\newline

If he randomly chose

6

exercises to do on Monday, what is the probability that all of them involve biking?

\newline

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

\newline

____

\newline

Total Exercises Count: Total number of exercises: $9$ $\newline$ Exercises involving biking: $7$ $\newline$ Exercises chosen for Monday: $6$ $\newline$ Calculate total possible combinations for choosing $6$ out of $9$ exercises.
Combination Formula: Use the combination formula: $^{n}C_{r} = \frac{n!}{r!(n-r)!}$ $\newline$ Total possible combinations: $^{9}C_{6}$
Total Combinations Calculation: ${}_9C_6 = \frac{9!}{6! \cdot (9-6)!}$ $= \frac{9!}{6! \cdot 3!}$ $= \frac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1}$ $= 84$
Biking Exercises Combinations: Calculate the combinations for choosing $6$ biking exercises out of $7$ . $\newline$ Favorable combinations: $_7C_6$
Biking Exercises Probability: $\binom{7}{6} = \frac{7!}{6! \cdot (7-6)!}$ $\newline$ $= \frac{7!}{6! \cdot 1!}$ $\newline$ $= 7$
Probability Calculation: Calculate the probability that all chosen exercises are biking. $\newline$ Probability = Favorable combinations / Total possible combinations $\newline$ = $\frac{7}{84}$
Fraction Simplification: Simplify the fraction to get the decimal. $\frac{7}{84} = \frac{1}{12}$

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