Integration to Solve Modified Sum of Binomial Coefficients
∑r =0r=nnrr+1
Question
r=n∑r=0(nr)r+1
A
2n
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B
2n+1−1n+1
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C
n
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D
nn
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Solution
The correct option is B2n+1−1n+1 (1+x)n=(n0)+(n1)x+....+(nn)xnIntegratingbothsidesfrom0to1∫10(1+x)ndx=∫10(n0)dx+∫10(n1)xdx+....+∫10(nn)xndx2n+1−1n+1=(n0)1+(n1)2+....+(nn)n+12n+1−1n+1=r=n∑r=0(nr)r+1