SATHEE: Permutations And Combinations Question 319

Permutations And Combinations Question 319

Question: The value of $ 1^{2}.C_1+3^{2}.C_3+5^{2}.C_5+… $ is:

Options:

A) $ n{{(n-1)}^{n-2+n{{.2}^{n-1}}}} $

B) $ n{{(n-1)}^{n-2}} $

C) $ n{{(n-1)}^{n-3}} $

D) None of these

Show Answer

Answer:

Correct Answer: D

Solution:

[d] We know, $ \sum\limits _{r=1}^{n}{r^{2}{{.}^{n}}C _{r}=n(n-1)}{2^{n-2}}+n{{.2}^{n-1}} $ ? (1) and $ \sum\limits _{r=1}^{n}{{{(-1)}^{r-1}}.r^{2}{{.}^{n}}C _{r}=0} $ ? (2) Adding (1) & (2) we get, $ 2[1^{2}.C_1+3^{2}.C_3+5^{2}C_5+…]=n(n-1){2^{n-2}}+n{{.2}^{n-1}} $

$ \Rightarrow [1^{2}C_1+3^{2}C_3+5^{2}C_5+…] $ $ =n(n-1){2^{n-3}}+n{{.2}^{n-2}}. $

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Permutations And Combinations Question 319

Question: The value of $ 1^{2}.C_1+3^{2}.C_3+5^{2}.C_5+… $ is:

Options:

Answer:

Solution:

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