The value of nC1+n+1C2+n+2C3+⋯+n+m−1Cm is equal to

nC1+n+1C2+n+2C3+…+n+m−1Cm =nCn−1+n+1Cn−1+n+2Cn−1+…+n+m−1Cn−1 = Coefficient of xn−1 in (1+x)n+(1+x)n+1+(1+x)n+2+…+(1+x)n+m−1 = Coefficient of xn−1 in (1+x)n(1+x)m−1−1(1+x)−1 = Coefficient of xn−1 in (1+x)m+n−(1+x)nx = Coefficient of xn in (1+x)m+n−(1+x)n =m+nCn-1Similarly, we can prove mC1+m+1C2+m+2C3+…+m+n−1Cn=m+nCm-1

The value of nC1+n+1C2+n+2C3+⋯+n+m−1Cm is equal to

nC1+n+1C2+n+2C3+…+n+m−1Cm =nCn−1+n+1Cn−1+n+2Cn−1+…+n+m−1Cn−1 = Coefficient of xn−1 in (1+x)n+(1+x)n+1+(1+x)n+2+…+(1+x)n+m−1 = Coefficient of xn−1 in (1+x)n(1+x)m−1−1(1+x)−1 = Coefficient of xn−1 in (1+x)m+n−(1+x)nx = Coefficient of xn in (1+x)m+n−(1+x)n =m+nCn-1Similarly, we can prove mC1+m+1C2+m+2C3+…+m+n−1Cn=m+nCm-1

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The value of $^{n} C_{1} +^{n + 1} C_{2} +^{n + 2} C_{3} + \dots +^{n + m - 1} C_{m}$ is equal to

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$^{m + n} C_{n} - 1$

$^{m + n} C_{n + 1}$

$^{m} C_{1} +^{m + 1} C_{2} +^{m + 2} C_{3} + \dots +^{m + n - 1} C_{n}$

$^{m + n} C_{m} - 1$

answer is A.

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Detailed Solution

$^{n} C_{1} +^{n + 1} C_{2} +^{n + 2} C_{3} + \dots +^{n + m - 1} C_{m} =^{n} C_{n - 1} +^{n + 1} C_{n - 1} +^{n + 2} C_{n - 1} + \dots +^{n + m - 1} C_{n - 1} = Coefficient of x^{n - 1} in (1 + x)^{n} + (1 + x)^{n + 1} + (1 + x)^{n + 2} + \dots + (1 + x)^{n + m - 1} = Coefficient of x^{n - 1} in (1 + x)^{n} [\frac{(1 + x)^{m - 1} - 1}{(1 + x) - 1}] = Coefficient of x^{n - 1} in \frac{(1 + x)^{m + n} - (1 + x)^{n}}{x} = Coefficient of x^{n} in [(1 + x)^{m + n} - (1 + x)^{n}] =^{m + n} C_{n} - 1$
Similarly, we can prove
$^{m} C_{1} +^{m + 1} C_{2} +^{m + 2} C_{3} + \dots +^{m + n - 1} C_{n} =^{m + n} C_{m} - 1$