How do you simplify $ ^7{P_5} $ ?.

Hint:In this question we need to simplify $ ^7{P_5} $ . Here, we will use the formula for permutations to simplify this. The formula of permutation is $ ^n{P_r} = \dfrac{{n!}}{{\left( {n - r}
\right)!}} $ . This refers to the arrangement of all the members of a set in some order or sequence.

Complete step-by-step solution:
Here, we need to simplify $ ^7{P_5} $ .
 $ ^7{P_5} $ is in the form of $ ^n{P_r} $ , therefore we can use the formula $ ^n{P_r} = \dfrac{{n!}}{{\left(
{n - r} \right)!}} $ .
Here $ ^n{P_r} $ represents $ n $ permutation $ r $ .
Where $ n $ is a set of things, and $ r $ is the arrangement of things where $ 0 < r \leqslant n $ .
The $ n! $ means the product of all positive integers less than or equal to $ n $ .
We can see from the given term $ ^7{P_5} $ in the place of $ n $ we have $ 7 $ , which shows that $ n = 7 $
Similarly, we can see that in the place of, we have $ 5 $ , which shows that $ r = 5 $ .
Therefore, substituting the value of $ n = 7 $ and $ r = 5 $ in the formula of the permutations, we have,
 $ ^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}} $
 $ ^7{P_5} = \dfrac{{7!}}{{\left( {7 - 5} \right)!}} $
 $ ^7{P_5} = \dfrac{{7!}}{{2!}} $
 $ ^7{P_5} = \dfrac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} $
 $ ^7{P_5} = 7 \times 6 \times 5 \times 4 \times 3 $
 $ ^7{P_5} = 2520 $

Hence, the permutation of $ ^7{P_5} $ is $ 2520 $ .

Note: It is important to note here that permutation refers to the process of arranging all the members of a given set to form a sequence without replacement. The number of permutations on a set of $ n $ elements is given by $ n! $ , where ‘ $ ! $ ’ represents factorials. When we come across the term permutation, we usually here, about the term combination. A combination is the choice of $ r $ things from a set of $ n $ things without replacement. The order does not matter in combination. It is given by the formula $ ^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $ . Permutations and combinations help us to determine the number of different ways of arranging and selecting objects without actually listening to them in real life.