द्विपद प्रमेय कि सहायता से `{(1+2x)^(6)(1-x)^(7)}` के प्रसार में `x^(5)` का गुणांक ज्ञात कीजिए ।

Explanation

`(1+2x)^(6)=sum_(r=0)^(6).^(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+160x^(2)+160x^(3)+240x^(4)+192x^(5)+64x^(6)`
और `(1-x)^(7)=sum_(r=0)^(7).^(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)`
अब `(1+2x)^(6)*(1-x)^(7)=(1+x)^(7)=(1+12x+60x^(2)+160x^(3)+240x^(4)+192x^(5)+64x^(6))*(1-7x+21x^(2)-35x^(3)+35x^(4)-21x^(5)+7x^(6)-x^(7))`
`thereforex^(5)` का गुणांक `=1xx(-21)+12xx35+60xx(-35)+160xx21+240xx(-7)+192xx1`
`=-21+420-2100+3360-1680+192`
=171.