Find the coefficient of `x^5`in the product `(1+2x)^6(1-x)^7`using binomial theorem.

Answer

N/a

Explanation

`(1+2x)^()=overset(6)underset(r=0)(Sigma).^(6)C_(r)(1)^(6-r).(2x)^(r)`
`-^(6)C_(0).(2x)^(0)+^(6)C_(1)(2x)^(1)+^(6)C_(2)(2x)^(2)`
`+^(6)C_(3)(2x)^(3)+^(6)C_(4)(2x)^(4)`
`+^(6)C_(5)(2x)^(5)+^(6)C_(6)(2x)^(6)`
`=1+12x_60x^(2)+160x^(3)+240x^(4)`
`_192x^(5)+64x^(6)`
`" and " (1-x)^(7) =overset(7)underset(r=0)(Sigma).^(7)C_(r)(1)^(7-r)(-x)^(r)`
`=^(7)C_(0)+^(7)C_(1).(-x)^(1)+^(7)C_(2)(-x)^(2)`
`+^(7)C_(3)(-x)^(3)+^(7)C_(4)(-x)^(4)+^(7)C_(5)(-x)^(5)`
`+^(7)C_(6)(-x)^(6)+^(7)C_(7)(-x)^(7)`
`=1-7x+21x^(2)-35x^(3)+35x^(4)`
`=21x^(5)+7x^(6)-x^(7)`
`"Now "(1+2x)^(6).(1-x)^(7)`
`=(1+12x+60x^(2)+160x^(3)+240x^(4)+192x^(3)`
`+64x^(6)).(1-7x+21x^(2)-35x^(3)`
`+35x^(4)-21x^(5)+7x^(6)-x^(7))`

`:. " Coefficient of " x^(5)=1xx (-21)+12xx35+60`
`xx(-35)+16)xx21+240xx(-7)+192xx1`
`=- 21 + 420 -2100+ 3360 -1680 + 192`
`=171`