Question

What is the value of \[^{7}{{P}_{2}}\]?

Answer 1

Hint: 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.