nCr Formula

Before going to learn the nCr formula, let us recall what is nCr. nCr is the number of ways of selecting some objects out of given objects where the order of the objects does not matter. It is expressed as \({}^n{C_r} = \frac{{{}^n{P_r}}}{{r!}} = \frac{{n!}}{{r!(n - r)!}}\). It is widely used in probability and statistics.

nCr formula has wide variety of applications in real life as well, like it can be used the number of ways of forming a team or a committee. Hence it plays vital rule in solving combinatorial problems. Let us learn the nCr formula along with a few solved examples.

What is the nCr Formula?

nCr formula is also known as the "combinations formula". nCr formula is used to find the number of ways of choosing r objects from n objects where 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.

☛ Note: \({}^n{C_r}\) is also written as nCr, nC_r, C(n, r), (or) \({}_n{C_r}\).

Derivation of nCr Formula

Let us recap, nPr (or P(n, r) formula, 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 fundamental counting principle, the number of ways to form a permutation is C(n, r) × r !. But this is nothing but permutation P(n, r). i.e., P(n, r) = C(n, r) × 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: Find the number of ways to select 3 books from 5 different books on the shelf.

Solution:

The total number of books, n = 5.

The number of books to be selected, r = 3.

The number of ways of selecting 3 books out of 5 books is 5C₃.

By nCr formula,

5C₃ = (5!) / ((5-3)! 3!)

= (120)/(2 × 6)

= 10

☛Check: You may check your answer for 5C₃ using the nCr Calculator.

Answer: The number of ways to select 3 books from 5 books is 10.

Example 2: Trevor has to choose 5 marbles from 12 different colored marbles. In how many ways can she choose them?

Solution:

Trevor has to choose 5 out of 12 marbles.

As order doesn't matter here, we use the nCr formula.

He can choose it in 12C₅ ways.

\(\begin{align}12C_5&= \dfrac{12!}{5 ! \times(12-5) !}\\ &= \dfrac{12!}{5 ! \times 7!}\\&= \dfrac{12\times 11\times 10\times 9\times 8\times 7 !}{5 ! \times 7 !}\\&= \dfrac{12\times 11\times 10\times 9\times 8}{5 \times4\times3\times2\times1 }\\&= 792\end{align}\)

Answer: In 792 ways she can choose the marbles.

Example 3: John asks his daughter to choose 4 pens from the basket. If the basket has 18 different pens to choose from, how many different possible ways she can do it?

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

Answer: The daughter can choose 4 pens from 18 pens in 3060 ways.