Question

How do you evaluate ${}^7{C}_2$?

Answer 1

Hint: The formula used to find the number of different combinations of n distinct elements taken r at a time is ${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}$ . By substituting the values from the question in the expression, we can find the value of ${}^7{C}_2$.

Complete Step by Step Solution:
We need to evaluate the given expression to find the possible combinations for choosing 2 elements from 7 distinct elements. We need to keep in mind that we need not consider the order in which these elements appear when chosen.
The formula we use to evaluate an expression of the form ${}^n{C}_r$ is
${\Rightarrow }^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}$
From the question, we have n=7 and r=2.
Substituting the values of n and r in the formula
$\Rightarrow {}^7{C}_2={\displaystyle \frac{7!}{2!(7-2)!}}={\displaystyle \frac{7!}{2!(5)!}}$
We further simply the factorial notations,
7! Can be expressed as $7!=7\times 6\times 5!$ . Therefore,
$\Rightarrow {}^7{C}_2={\displaystyle \frac{7\times 6\times 5!}{(2\times 1)(5)!}}$
Cancelling common terms in the above expression, we get
$\Rightarrow {}^7{C}_2={\displaystyle \frac{7\times 6}{2\times 1}}$
Simplifying the expression further,
$\Rightarrow {}^7{C}_2={\displaystyle \frac{42}{2}}=21$

Hence, the value of the expression ${}^7{C}_2$ is 21. This means that there are 21 combinations for choosing 2 elements from 7 distinct elements.

Note:
The factorial function which is represented by the symbol “!” simply means that we need to multiply a series of descending natural numbers. For example, $4!=4\times 3\times 2\times 1$. The value of 0! Is considered to be 1.