nCr + n-1Cr + n-2Cr +............+ n-[n-r]Cr =coef. of x^r in [1+x]n +coef. of x^r in [1+x]n-1 +coef. of x^r in [1+x]n-2 +..........+coef. of x^r in [1+x]n-(n-r)
=coef. of x^r in {[1+x]n +[1+x]n-1+[1+x]n-2+........+[1+x]n-(n-r) }
=coef. of x^r in (1+x)n { 1+1/[1+x] +1/[1+x]2 + 1/[1+x]3 +....+1/[1+x]n-r}
=coef. of x^r in (1+x)n {( 1- 1/[1+x]n-r+1)/(1-1/[1+x])}
=coef. of x^r in (1+x)n {( [1+x]n-r+1 -1)/(x[1+x]n-r)}
=coef. of x^r in (1+x)n-(n-r) {([1+x]n-r+1-1)/x}
=coef. of x^r+1 in (1+x)r{[1+x]n-r+1-1}
=coef. of x^r+1 in (1+x)n+1-(1+x)r
=n+1Cr+1 + 0 (since max. coeff. of x in (1+x)r is r)
Moderation Team
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