NCERT Solutions
Class 11 Maths
Binomial Theorem

Ex.Misc.Q.3
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
Using Binomial Theorem, the expressions, (1 + 2x)6 and (1 – x)7, can be expanded as
(1 + 2x)6 = 6C0 + 6C1 (2x) + 6C2 (2x)2 + 6C3 (2x)3 + 6C4 (2x)4 + 6C5 (2x)5 + 6C6 (2x)6
= 1 + 6(2x) + 15(2x)2 + 20(2x)3 + 15(2x)4 + 6(2x)5 + (2x)6
= 1 + 12x + 60x2 + 160x3 + 240x4 + 192x5 + 64x6
(1 - x)7 = 7C0 - 7C1 (x) + 7C2 (x)2 - 7C3 (x)3 + 7C4 (x)4 - 7C5 (x)5 + 7C6 (x)6 - 7C7 (x)7
= 1 - 7x + 21x2 - 35x3 + 35x4 - 21x5 + 7x6 - x7
Now, (1 + 2x)6 and (1 – x)7
= (1 + 12x + 60x2 + 160x3 + 240x4 + 192x5 + 64x6) * (1 - 7x + 21x2 - 35x3 + 35x4 - 21x5 + 7x6 - x7)
The complete multiplication of the two brackets is not required to be carried out.
Only those terms, which involve x5, are required.
The terms containing x5 are
1(-21x5) + (-12x) (35x4) + (60x2) (-35x3) + (160x3) (21x2) + (240x4) (-7x) + (192x5) (1) = 171x5
Thus, the coefficient of x5 in the given product is 171.