S1-mc1-m-1C2-m-2C3-m-n-1Cn S2-nc1-n-1C2-n-2C3-m-n-1Cn

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Sum of Coefficients of All Terms

S1=mc1+m+1C2+...

Question

$S_{1} =^{m} c_{1} + (m + 1)_{C_{2} +} (m + 2)_{C_{3} + \dots . +} (m + n - 1)_{C_{n}}$
$S_{2} =^{n} c_{1} + (n + 1)_{C_{2} +} (n + 2)_{C_{3} + \dots . +} (m + n - 1)_{C_{n}}$

S_{1} + S_{2} = 0

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S_{1} - S_{2} = 0

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S_{1} + S_{2} = 2^{n}

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S_{1} + S_{2} = 2^{n} - 1

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Solution

The correct option is A $S_{1} - S_{2} = 0$
$S_{1}$ = $^{n} C_{1}$ + $^{m + 1} C_{2}$ + $^{m + 2} C_{3}$ +........+ $^{m + n - 1} C_{n}$
Adding 1 both sides
$S_{1}$ +1= $^{m} C_{0}$ + $^{m} C_{1}$ ........... using ( $^{n} C_{r - 1}$ + $^{n} C_{r}$ = $^{n + 1} C_{r}$ )
$S_{1}$ +1= $^{m + 1} C_{1}$ + $^{m + 1} C_{2}$ .........
Similarly, this can seen that $S_{1}$ +1= $^{m + n - 1} C_{n - 1}$ + $^{m + n - 1} C_{n}$ = $^{m + n} C_{n}$ eqn 1

Similarly,it can be seen that $S_{2}$ +1= $^{m + n} C_{n}$ eqn 2
(eqn 1)-(eqn 2)

( $S_{1}$ +1)-( $S_{2}$ +1)= $^{m + n} C_{n}$ - $^{m + n} C_{n}$
$S_{1}$ - $S_{2}$ =0

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S1=mc1+(m+1)C2+(m+2)C3+….+(m+n−1)CnS2=nc1+(n+1)C2+(n+2)C3+….+(m+n−1)Cn

$S_{1} =^{m} c_{1} + (m + 1)_{C_{2} +} (m + 2)_{C_{3} + \dots . +} (m + n - 1)_{C_{n}}$
$S_{2} =^{n} c_{1} + (n + 1)_{C_{2} +} (n + 2)_{C_{3} + \dots . +} (m + n - 1)_{C_{n}}$