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Show that nC0.nC0 - n+1C1.nC1 + n+2C2.nC2-....=(-1)n. |
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Answer» nC0.nC0 - n+1C1.nC1 + n+2C2.nC2 - .... is coefficient of xn in (nC0(x + 1)n - nC1(x + 1)n+1 + nC2(x + 1)n+2-.....) Now, nC0 (x + 1)n - nC1(x + 1)n+1 + nC2(x + 1)2-..... = (x + 1)n (nC0 - nC1(x + 1) + nC2(x + 1)2 -....) = (x + 1)n (1 - (x + 1)n) (\(\because\) (1 - x)n = nC0- nC1x + nC2x2-....) = (-x)n (x + 1)n = (-1)n xn (x +1)n Hence, Coefficient of xn in (nC0(x + 1)n - nC1(x + 1)n+1 + nC2(x+1)2) is (-1)n \(\therefore\) nC0.nC0 - n+1C1.nC1 + n+2C2.nC2-....=(-1)n |
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