यदि sum_{k=1}^{30} k left({ }^{30} C _kright) | vedclass

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(C) माना $S = \sum_{k=1}^{30} k \left({ }^{30} C _k\right)^2$. गुणधर्म ${ }^{n} C _k = { }^{n} C _{n-k}$ का उपयोग करने पर,$S = \sum_{k=1}^{30} k \left

Vedclass · Accepted Answer

$15$

Vedclass · Answer

(C) माना $S = \sum_{k=1}^{30} k \left({ }^{30} C _kight)^2$. गुणधर्म ${ }^{n} C _k = { }^{n} C _{n-k}$ का उपयोग करने पर,$S = \sum_{k=1}^{30} k \left({ }^{30} C _{30-k}ight)^2$. $j = 30-k$ लेने पर,$S = \sum_{j=0}^{29} (30-j) \left({ }^{30} C _jight)^2$. अतः,$2S = 30 \sum_{k=0}^{30} \left({ }^{30} C _kight)^2 = 30 	imes { }^{60} C _{30}$. $S = 15 	imes { }^{60} C _{30} = 15 	imes \frac{60!}{(30!)^2}$. इसलिए,$\alpha = 15$.

यदि $\sum_{k=1}^{30} k \left({ }^{30} C _k\right)^2 = \frac{\alpha 60 !}{(30 !)^2}$ है,तो $\alpha$ का मान ज्ञात कीजिए।

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