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.

Derivation of NCR Formula

Let us recap, \(P(n, r)\), the number of ways to form a permutation of \(r\) elements from a total of \(n\) can be determined by:
1. Forming a combination of \(r\) elements out of a total of \(n\) in any one of \(C(n, r)\) ways
2. Ordering these \(r\) elements any one of \(r !\) ways.
By the multiplication principle, the number of ways to form a permutation is \(P(n, r)=\) \(C(n, r) \times r !\)
Using the formula for permutations \(P(n, r)=n ! /(n-r) !\) to substitute into the above formula:
\(n ! /(n-r) !=C(n, r) r !\)
On solving this, the number of combinations, \(C(n, r)=n ! /[r !(n-r) !]\)

Let us check out a few solved examples to understand more about NCR formulas.

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.

\[\begin{align}12C_5&= \dfrac{12!}{5 ! \times(12-5) !}\\ &= \dfrac{12!}{5 ! \times 7!}\\&= \dfrac{12\times 11\times 10\times 9\times 8\times 7 !}{5 ! \times 7 !}\\&= \dfrac{12\times 11\times 10\times 9\times 8}{5 ! }\\&= 792\end{align}\]

Answer: In 792 ways she can choose the marbles.

Example 3: John asks his daughter to choose 4 apples from the basket. If the basket has 18 apples to choose from, how many different possible answers could the daughter give?

Solution: 

Given,
r = 4 (sub-set)
n = 18

Therefore, we need to find “18 Choose 4”

Now, Combination = C(n, r) = n!/r!(n–r)!
18!/4!(18−4)!=18!/14!×4!

= 3,060 possible answers.

Answer: The daughter can give 3060 possible answers.