If ( C 1 30 ) 2 + 2 ( C 2 30 ) 2 + 3 ( C 3 30 ) 2 + . . . + 30 ( C 30 30 ) 2 = 60 ! ( 30 ! ) 2 , then is equal to [2023]

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Solution

Q.
If ${({}^{30}{C_{1}})}^{2} + 2 {({}^{30}{C_{2}})}^{2} + 3 {({}^{30}{C_{3}})}^{2} + . . . + 30 {({}^{30}{C_{30}})}^{2} = \frac{α 60!}{{(30!)}^{2}},$ then $α$ is equal to [2023]

1 60

2 10

3 15

4 30

Ans.
(3)

$Given,$ ${({}^{30}C_{1})}^{2} + 2 {({}^{30}C_{2})}^{2} + 3 {({}^{30}C_{3})}^{2} + \dots + 30 {({}^{30}C_{30})}^{2} = \frac{α 60!}{{(30!)}^{2}}$

$\sum_{r = 1}^{30} r \cdot {({}^{30}C_{r})}^{2}$ $= \sum_{r = 1}^{30} r \cdot {}^{30}C_{r} \cdot {}^{30}C_{r}$

$= \sum_{r = 1}^{30} r \cdot \frac{30}{r} \cdot {}^{29}C_{r - 1} \cdot {}^{30}C_{r}$ $= \sum_{r = 1}^{30} 30 \cdot {}^{29}C_{r - 1} {}^{30}C_{30 - r}$ $= 30 {}^{59}C_{30}$

$\Rightarrow 30 \frac{59!}{30! 29!} \frac{30}{30} = \frac{15 \cdot 60!}{{(30)}^{2}} ∴ α = 15$

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