Binomial Theorem Formula

If you want to expand a binomial expression with some higher power, then Binomial theorem formula works well for it. Following is the Binomial theorem formula:

(x + y)n = Σr=0n nCr xn – r · yr

where, nCr = n!⁄(n-r)!r!

Where n! Denotes the product of all the whole numbers between 1 to n.

For Example 5! = 1 x 2 x 3 x 4 x 5.

Binomial Theorem Example

Q.1: Find the value of 10C6.

Solution:

 10C6 = 10 ! / (10 – 6)! 6! = 10! / 4! 6!

= (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10) /( 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 5 x 6)

= (7 x 8 x 9 x 10) /(1 x 2 x 3 x 4)

= 7 x 3 x 10

= 210

Q.2: Expand (x2 + 2)6

Solution:
(x2 + 2)6 = 6C (x2)6(2)0 + 6C1(x2)5(2)1 + 6C2(x2)4(2)2 + 6C (x2)3(2)3 + 6C (x2)2(2)4 + 6C (x2)1(2)5 + 6C6 (x2)0(2)6

= (1)(x12)(1) + (6)(x10)(2) + (15)(x8)(4) + (20)(x6)(8) + (15)(x4)(16) + (6)(x2)(32) + (1)(1)(64)

= x12 + 12 x10 + 60 x8 + 160 x6 + 240 x4 + 192 x2 + 64

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