NCR Formula
Before going to learn the NCR formula, let us recall what is NCR. NCR is a combination and it is an arrangement in which the order of the objects does not matter. NCR is known as the selection of things without considering the order of the arrangement. NCR formula is used to find the possible arrangements where selection is done without order consideration. Let us learn the NCR formula along with a few solved examples.
What Is the NCR Formula?
NCR formula is used to find the number of ways where r objects chosen from n objects and the order is not important. It is represented in the following way.
\({}^n{C_r} = \frac{{{}^n{P_r}}}{{r!}} = \frac{{n!}}{{r!(n  r)!}}\)
Here,
 n is the total number of things.
 r is the number of things to be chosen out of n things.
Let us learn the NCR formula along with a few solved examples below.
Solved Examples Using NCR Formula

Example 1: Use the NCR formula to find the number of ways to select 3 books from 5 books on the shelf.
Solution
We can choose it in \(5P_3 = 60 \) ways.
The possible ways the books could be selected doesn't require order as we can choose any at random.
Thus we divide 60 by 3 !. i.e. 60/6 = 10 ways.
Answer: The number of ways to select 3 books from 5 books is 10.

Example 2: Trevor has to choose 5 marbles from 12 marbles. In how many ways can she choose them?
Solution
Patricia has to choose 5 out of 12 marbles.
As order doesn't matter here, we use the NCR formula.
Thus he can choose it in \(12C_5\) ways.
\[\begin{align}12C_5&= \dfrac{12!}{5 ! \times(125) !}\\ &= \dfrac{12!}{5 ! \times 7!}\\&= \dfrac{12\times 11\times 10\times 9\times 8\times 7 !}{5 ! \times 7 !}\\&= \dfrac{12\times 11\times 10\times 9\times 8}{5 ! }\\&= 792\end{align}\]
Answer: In 792 ways she can choose the marbles.