Question

How do you evaluate ${}^{5}{{p}_{5}}$?

Answer 1

To solve this problem, we will use the formula of permutation. ${}^{n}{{p}_{r}}=\dfrac{n!}{(n-r)!}$ is the formula of permutation which will help us to solve this problem. In the formula where n is the number of permutations within r time is determined by ${}^{n}{{p}_{r}}=\dfrac{n!}{(n-r)!}$. So, let’s see how we can solve this problem.

Complete Step by Step Solution:
In the given question we have to evaluate ${}^{5}{{p}_{5}}$ . So here we will use the formula of permutation that is ${}^{n}{{p}_{r}}=\dfrac{n!}{(n-r)!}$ where $n! = n (n-1) (n-2)…… \times 3 \times 2 \times 1$.
So here the value of n is 5 and the value of r is 5.
$\Rightarrow {}^{5}{{p}_{5}}=\dfrac{5!}{(5-5)!}$
$\Rightarrow {}^{5}{{p}_{5}}=\dfrac{5!}{0!}$
So here we will solve the factorial
$\Rightarrow \dfrac{5!}{0!}=\dfrac{5\times 4\times 3\times 2\times 1}{1}$ [where $\;0!$ expands to 1 and $\;5!$ expands to 120]
$=\dfrac{120}{1}$
Now, we will calculate the numerator and denominator
= 120

$\therefore {}^{5}{{p}_{5}}=120$

Additional Information:
The act of arranging the objects or numbers in some specific sequence or order is known as permutation. And the combination is the way of selecting objects such that the selection order does not affect the combination.

Note:
In the given problem the factorial “!” is the important part. For solving the factorial, we have kept subtracting and multiply the number until we reach 1. Like if we have a number 10 and we have to find the factorial of 10. Then 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800 will be the factorial of 10. So, as you have seen that we will stop once we reach 1.