Answer :

We need to evaluate the permutation

[tex]$$
{}_5P_5 = \frac{5!}{(5-5)!}.
$$[/tex]

Step 1: Calculate the factorial in the numerator. By definition,

[tex]$$
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120.
$$[/tex]

Step 2: Calculate the factorial in the denominator. Since

[tex]$$
(5-5)! = 0!,
$$[/tex]

and by definition,

[tex]$$
0! = 1,
$$[/tex]

we have

[tex]$$
(5-5)! = 1.
$$[/tex]

Step 3: Substitute these values into the permutation formula:

[tex]$$
{}_5P_5 = \frac{5!}{(5-5)!} = \frac{120}{1} = 120.
$$[/tex]

Thus, the value of [tex]${}_5P_5$[/tex] is [tex]$\boxed{120}$[/tex].

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