Question
Evaluate the expression \( { }_{7} \mathrm{P}_{2} \) \( { }_{7} \mathrm{P}_{2}=\square \)
Ask by Simpson Munoz. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(_{7}P_{2} = 42\)
Solution
Given the formula for permutations:
\[
{}_{n}P_{r} = \frac{n!}{(n-r)!}
\]
Substitute \( n = 7 \) and \( r = 2 \):
\[
{}_{7}P_{2} = \frac{7!}{(7-2)!} = \frac{7!}{5!}
\]
Simplify by expanding the factorial:
\[
\frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 = 42
\]
Thus, the evaluated expression is:
\[
{}_{7}P_{2} = 42
\]
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Simplify this solution The Deep Dive
To evaluate the expression \( { }_{7} \mathrm{P}_{2} \), we use the formula for permutations, which is given by: \[ _{n}P_{r} = \frac{n!}{(n-r)!} \] In this case, \( n = 7 \) and \( r = 2 \). Plugging in these values, we get: \[ _{7}P_{2} = \frac{7!}{(7-2)!} = \frac{7!}{5!} \] Now, we can simplify \( \frac{7!}{5!} \): \[ 7! = 7 \times 6 \times 5! \] So, \[ \frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 \] Calculating that gives: \[ 7 \times 6 = 42 \] Thus, \[ _{7}P_{2} = 42 \] So the final answer is \( { }_{7} \mathrm{P}_{2} = 42 \).
