Find the coefficient of x5 in the product (1+2x)6(1−x)7 using binomial theorem.
By using the formula :
(a+b)n=nC0an+nC1an−1b+nC2an−2b2+.........nCran−rbr+......nCnbn
(1+2x)6=6C0.1+6C1(2x)+6C2(2x)2+6C2(2x)3+6C4(2x)4+6C6(2x)6
=1+12x+60x2+20×8x3+15×16x4+6×32x6+64x5
=1+12x+60x2+160x3+240x4+192x5+64x6------------(i)
(1−x)7=1−7C1x+7C2x2−7C3x3+7C4x4−7C5x5+....7C7x7
=1−7x+21x2−35x3+35x4−21x5+7x6−x7-----------(ii)
multiply equation (i) and (ii) .
=1×(−21)+12×35+60×(−35)+160×21+240×(−7)+192×1
=−21+420−2100+3360−1680+192
=171