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Question

If (1+x)n=nC0+nC1x+nC2x2+nCnxn,xϵN then prove that

nC0nC1+nC2......+(1)m1×nCm1

=(1)m1(n1)(n2)..(nm+1)(m1)!

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Solution

R.H.S.=(1)m1(n1)(n2)....(nm+1)(m1)!(nm)(nm1)....2.1(nm)(nm1)....2.1

=(1)m1(n1)!(m1)!(nm)!

=(1)m1×n1Cm1

Now (1)m1×n1Cm1= coeff. of xm1 in the
or in (1x)n(1x)1

or in [C0C1x+C2x2....+(1)(m1)Cm1xm1....][1+x+x2+....+xm1+....]
Coeff of xm1 in the above product is clearly

C0C1+C2....+(1)m1Cm1

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