Binomial Coefficient Formula

Binomial coefficient is a mathematical term used in binomial expansion. In shortly, the binomial coefficient is represented as `((n),(r))`. In binomial expansion the coefficients are termed as binomial coefficients. There is a general formula is available for binomial coefficients. This article gives the formula used in binomial coefficient and some example problems using that formula.

 

Expansion to Binomial Coefficient Formula:

 

Representations of binomial coefficients:

  • `((n),(r))`
  • nCr
  • C(n,r)

We can solve this binomial coefficient by using the given formula.

`((n),(r))` = `(n!)/(r!(n-r)!)`

This formula is same as the combination formula.

 

Example Problems to Binomial Coefficient Formula:

 

Example: 1

Solve the binomial coefficient: `((4),(1))`

Solution:

Given binomial coefficient is `((4),(1))`

Step 1:

The general formula of binomial coefficient is,

`((n),(r))` = `(n!)/(r!(n-r)!)`

Step 2:

`((4),(1))``(4!)/((1!)(4-1)!)`

Step 2:

4! = 1 .2 . 3 . 4

Step 3:

1! = 1

(4 - 1)! = 3!

  = 1 .2 . 3

Step 4:

`((4), (1))` = `(1 . 2 . 3 . 4)/((1)(1 . 2 . 3))`

Cancel the common terms

`4/1`

= 4

Answer: `((4),(1))` = 4

 

Example: 2

Solve the binomial coefficient: `((9),(2))`

Solution:

Given binomial coefficient is `((9),(2))`

Step 1:

The general formula of binomial coefficient is,

`((n),(r))` = `(n!)/(r!(n-r)!)`

Step 2:

`((9),(2))``(9!)/((2!)(9-2)!)`

Step 3:

9! = 1 .2 . 3 . 4 . 5 . 6. 7. 8. 9

Step 4:

2! = 1 . 2

(9 - 2)! = 7!

   = 1. 2 . 3. 4. 5 . 6 . 7

Step 5:

`((9), (2))` = `(1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9)/((1 . 2)(1 . 2 . 3 . 4 . 5 . 6 . 7))`

Cancel the common terms

`(8 . 9)/2`

= 4 x 9

= 36

Answer: `((9),(2))` = 36

 

Example: 3

Solve the binomial coefficient: `((6),(3))`

Solution:

Given binomial coefficient is `((6),(3))`

Step 1:

The general formula of binomial coefficient is,

`((n),(r))` = `(n!)/(r!(n-r)!)`

Step 2:

`((6),(3))``(6!)/((3!)(6-3)!)`

Step 3:

6! = 1 .2 . 3 . 4 . 5 . 6

Step 4:

3! = 1 . 2 . 3

(6 - 3)! = 3!

  = 1 . 2 . 3

Step 4:

`((6), (3))` = `(1 . 2 . 3 . 4 . 5 . 6)/((1 . 2 . 3)(1 . 2 . 3))`

Cancel the common terms

`(4 . 5 . 6)/(1 . 2 . 3)`

= 4 x 5

= 20

Answer: `((6),(3))` = 20

 

Practice Problems to Binomial Coefficient Formula:

 

Problem: 1

Solve binomial coefficient:

`((5),(4))`

Answer: `((5),(4))`= 5

Problem: 2

Solve binomial coefficient:

`((3),(1))`

Answer: `((3),(1))`= 3