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

Answer

N/a

Explanation

`(1+2x)^(6)=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_(5)(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+160x^(2)+160x^(3)`
`+240x^(4)|192x^(5)|64x^(6)).(1" "7x|21x^(2)`
`-35x^(3)+35x^(4)-21x^(5)+7x^(6)-x^(7))`
`:."Coefficient of "x^(5)=1xx(-21)+12xx35+60xx(-35)`
`+160xx21+240xx(-7)+192xx1`
`=-21+420-2100+3360`
`-1680+192=171`