Question

How do you evaluate ${}^{9}{{P}_{7}}$ ?

Answer 1

Hint: The value of ${}^{n}{{P}_{x}}$ is equal to product of ${}^{n}{{C}_{x}}$ and $x!$ where $x!$ is the product of all the numbers from 1 to x . We know that the value of ${}^{n}{{C}_{x}}$ is equal to $\dfrac{n!}{x!\left( n-x \right)!}$ , so ${}^{n}{{P}_{x}}$ is equal to $\dfrac{n!}{\left( n-x \right)!}$ . By using this formula we can calculate the value of ${}^{9}{{P}_{7}}$

Complete step-by-step answer:
We have to evaluate ${}^{9}{{P}_{7}}$
We know that ${}^{9}{{P}_{7}}$ is equal to product of ${}^{n}{{C}_{x}}$ and $x!$ which is $\dfrac{n!}{\left( n-x \right)!}$
So we can write ${}^{9}{{P}_{7}}$ as $\dfrac{9!}{\left( 9-7 \right)!}$ which is $\dfrac{9!}{2!}$
The value of 9 factorial is equal to product of all numbers from 1 to 9 and the value of 2 factorial is equal to product of all number from 1 to 2
So we can write $9!=1\times 2\times ....\times 9$ which is equal to 362880
 And factorial of 2 is equal to 2
We can write $\dfrac{9!}{2!}$ = $\dfrac{362880}{2}$ which is 181440

Note: ${}^{n}{{C}_{x}}$ denotes the total number of possible combination of x different object out of n different objects and ${}^{n}{{P}_{x}}$ denotes the total number of ways we can arrange x different objects out of n different objects for example the number of ways to select 4 digits out of 9 is equal to ${}^{9}{{C}_{4}}$ and the number of ways to arrange 4 digits out of 9 is equal to ${}^{9}{{P}_{4}}$. In the combination we do not have to arrange for example 1, 2, 3 and 1, 3, 2 are the same in combination but they are different in permutation.