Question

What is the value of $^{7}{{P}_{2}}$?

Answer 1

In this problem, we have to evaluate the given permutation. We know that permutation is defined as the arrangement of elements into a sequence or a linear order. The permutation means selection of things with the importance of order. It is denoted as $^{n}{{P}_{r}}$. Here we are given an expression, to find its permutation. We can use the permutation formula $^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$. We can substitute the given values in the formula to find the answer.

Complete step-by-step solution:
We know that the given expression to be evaluated is,
$^{7}{{P}_{2}}$
We know that permutation is defined as the arrangement of elements into a sequence or a linear order. The permutation means selection of things with the importance of order. It is denoted as $^{n}{{P}_{r}}$.
We know that the permutation formula that can be used to evaluate is,
$^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$
Where, $^{n}{{P}_{r}}$ is the permutation, n is the total number of objects in the set, r is number of selected objects in a set.
From the given expression, we can see that, n = 7, r = 2.
We can now substitute the above values in the permutation formula, we get
${{\Rightarrow }^{7}}{{P}_{2}}=\dfrac{7!}{\left( 7-2 \right)!}=\dfrac{7\times 6\times 5\times 4\times 3\times 2\times 1}{5\times 4\times 3\times 2\times 1}$
We can now simplify by cancelling the similar terms in the above step, we get
${{\Rightarrow }^{7}}{{P}_{2}}=7\times 6=42$
Therefore, the answer is 42.

Note: We should remember that the formula for permutation is $^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$, Where, $^{n}{{P}_{r}}$ is the permutation, n is the total number of objects in the set, r is number of selected objects in a set. We should also know that factorial is a function that multiplies a number by every number below it.