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Question
If (1+x)n=nC0+nC1x+nC2x2+nCnxn,xϵN then prove that
nC0−nC1+nC2−......+(−1)m−1×nCm−1
=(−1)m−1(n−1)(n−2)..(n−m+1)(m−1)!
nC0−nC1+nC2−......+(−1)m−1×nCm−1
=(−1)m−1(n−1)(n−2)..(n−m+1)(m−1)!
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Solution
R.H.S.=(−1)m−1(n−1)(n−2)....(n−m+1)(m−1)!⋅(n−m)(n−m−1)....2.1(n−m)(n−m−1)....2.1
=(−1)m−1(n−1)!(m−1)!(n−m)!
=(−1)m−1×n−1Cm−1
Now (−1)m−1×n−1Cm−1= coeff. of xm−1 in the
or in (1−x)n(1−x)−1
or in [C0−C1x+C2x2−....+(−1)(m−1)Cm−1xm−1....]⋅[1+x+x2+....+xm−1+....]
Coeff of xm−1 in the above product is clearly
C0−C1+C2−....+(−1)m−1Cm−1
=(−1)m−1(n−1)!(m−1)!(n−m)!
=(−1)m−1×n−1Cm−1
Now (−1)m−1×n−1Cm−1= coeff. of xm−1 in the
or in (1−x)n(1−x)−1
or in [C0−C1x+C2x2−....+(−1)(m−1)Cm−1xm−1....]⋅[1+x+x2+....+xm−1+....]
Coeff of xm−1 in the above product is clearly
C0−C1+C2−....+(−1)m−1Cm−1
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