Question
Evaluate the expression \( { }_{8} \mathrm{P}_{2} \) \( { }_{8} \mathrm{P}_{2}=\square \)
Ask by Bowers Johnston. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(_{8} \mathrm{P}_{2} = 56\)
Solution
We start with the definition of permutations:
\[
{}_{n}\mathrm{P}_{r}=\frac{n!}{(n-r)!}
\]
Substituting \( n=8 \) and \( r=2 \):
\[
{}_{8}\mathrm{P}_{2}=\frac{8!}{(8-2)!}=\frac{8!}{6!}
\]
Now, we simplify the factorial expression:
\[
8! = 8 \times 7 \times 6!
\]
Thus:
\[
{}_{8}\mathrm{P}_{2}=\frac{8 \times 7 \times 6!}{6!}=8 \times 7=56
\]
The final evaluation is:
\[
{}_{8}\mathrm{P}_{2}=56
\]
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Simplify this solution Mind Expander
To evaluate the expression \( { }_{8} \mathrm{P}_{2} \), we use the formula for permutations, which is given by: \[ _{n}P_{r} = \frac{n!}{(n-r)!} \] In this case, \( n = 8 \) and \( r = 2 \). Plugging in these values gives us: \[ _{8}P_{2} = \frac{8!}{(8-2)!} = \frac{8!}{6!} \] Now, we can simplify \( \frac{8!}{6!} \) by noting that \( 8! = 8 \times 7 \times 6! \): \[ _{8}P_{2} = \frac{8 \times 7 \times 6!}{6!} \] The \( 6! \) in the numerator and denominator cancels out: \[ _{8}P_{2} = 8 \times 7 = 56 \] Thus, \[ { }_{8} \mathrm{P}_{2} = 56 \]
