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  4. If ( 30 C 1)2+2( 30 C 2)2+3( 30 C 3)2+ ldots+30( 30 C 30)2=(α 60 !/(30 !)2) then α is equal to :

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Q. If $\left({ }^{30} C _1\right)^2+2\left({ }^{30} C _2\right)^2+3\left({ }^{30} C _3\right)^2+\ldots+30\left({ }^{30} C _{30}\right)^2=\frac{\alpha 60 !}{(30 !)^2}$ then $\alpha$ is equal to :

JEE MainJEE Main 2023Permutations and Combinations

Solution:

$S =0 \cdot\left({ }^{30} C _0\right)^2+1 \cdot\left({ }^{30} C _1\right)^2+2 \cdot \cdot\left({ }^{30} C _2\right)^2+\ldots \ldots+30 \cdot\left({ }^{30} C _{30}\right)^2 $
$S =30 .\left({ }^{30} C _0\right)^2+29 .\left({ }^{30} C _1\right)^2+28 .\left(^{30} C _2\right)^2+\ldots . .+0 .\left({ }^{30} C _0\right)^2$
$ 2 S =30 \cdot\left({ }^{30} C _0{ }^2++^{30} C _1^2+\ldots \ldots . \cdot{ }^{30} C _{30}{ }^2\right) $
$ S =15 \cdot{ }^{60} C _{30}=15 \cdot \frac{60 !}{(30 !)^2} $
$ \frac{15 \cdot 10 !}{(30 !)^2}=\frac{\alpha \cdot 60 !}{(30 !)^2}$
$\Rightarrow \alpha=15$