Question

How do you evaluate $5\left( {}^{3}{{P}_{2}} \right)$?

Answer 1

Hint: Here we need to find the value of the following expression. We will use the formula of permutation here. Then after applying the formula of permutation, we will find the value of the factorials and we will substitute their values in the given expression and then we will apply certain mathematical operation will be required to solve the expression like subtraction, addition, and division to simplify the expression further and hence to get the required value of the expression.

Formula used:
The formula of permutation is given by ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$.

Complete step by step solution:
Here we need to find the value of the following expression and the given expression is $5\left( {}^{3}{{P}_{2}} \right)$.
Using the formula of permutation in the given expression, we get
$ 5\left( {}^{3}{{P}_{2}} \right)=5\times \dfrac{3!}{\left( 3-2 \right)!}$
Now, we will subtract 2 from 3 in the denominator. Therefore, we get
$ \Rightarrow 5\left( {}^{3}{{P}_{2}} \right)=5\times \dfrac{3!}{1!}$
Now, we will substitute the value of the factorials of the numbers 3 and 1.
$\Rightarrow 5\left( {}^{3}{{P}_{2}} \right) =5\times \dfrac{3\times 2\times 1}{1}$
On multiplying the numbers, we get
$ \Rightarrow 5\left( {}^{3}{{P}_{2}} \right) =30$

Therefore, the value of the given expression is equal to 30 i.e. value of $5\left( {}^{3}{{P}_{2}} \right)$ is equal to 30.

Note:
Here we have obtained the value of the given expression. We have used the formula of permutation to simplify it. A permutation is defined as the mathematical technique which determines the number of possible arrangements in a set when the order of the arrangements always matters. Students get confused between the terms permutation and combination. A combination is defined as the mathematical technique which determines the number of possible arrangements in a set where the order of the selection does not matter.