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Answer :
Sure! Let's evaluate [tex]\( {}_5 P_5 \)[/tex].
The expression [tex]\( {}_n P_r \)[/tex] represents the number of permutations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time. The formula to calculate permutations is:
[tex]\[
{}_n P_r = \frac{n!}{(n-r)!}
\][/tex]
For the given problem, we have [tex]\( n = 5 \)[/tex] and [tex]\( r = 5 \)[/tex].
Let's substitute these values into the formula:
1. Calculate [tex]\( 5! \)[/tex] (which means 5 factorial):
[tex]\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\][/tex]
2. Calculate [tex]\( (5-5)! = 0! \)[/tex]:
[tex]\[
0! = 1
\][/tex]
(By definition, the factorial of 0 is 1.)
3. Substitute these values back into the permutation formula:
[tex]\[
{}_5 P_5 = \frac{5!}{0!} = \frac{120}{1} = 120
\][/tex]
So, the evaluation of [tex]\( {}_5 P_5 \)[/tex] is 120.
The expression [tex]\( {}_n P_r \)[/tex] represents the number of permutations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time. The formula to calculate permutations is:
[tex]\[
{}_n P_r = \frac{n!}{(n-r)!}
\][/tex]
For the given problem, we have [tex]\( n = 5 \)[/tex] and [tex]\( r = 5 \)[/tex].
Let's substitute these values into the formula:
1. Calculate [tex]\( 5! \)[/tex] (which means 5 factorial):
[tex]\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\][/tex]
2. Calculate [tex]\( (5-5)! = 0! \)[/tex]:
[tex]\[
0! = 1
\][/tex]
(By definition, the factorial of 0 is 1.)
3. Substitute these values back into the permutation formula:
[tex]\[
{}_5 P_5 = \frac{5!}{0!} = \frac{120}{1} = 120
\][/tex]
So, the evaluation of [tex]\( {}_5 P_5 \)[/tex] is 120.
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