Question:
If $ n=5, $ then $ {{{{(}^{n}}{{C}_{0}})}^{2}}+{{{{(}^{n}}{{C}_{1}})}^{2}}+{{{{(}^{n}}{{C}_{2}})}^{2}}+..... $ $ +{{{{(}^{n}}{{C}_{5}})}^{2}} $ is equal to
If $ n=5, $ then $ {{{{(}^{n}}{{C}_{0}})}^{2}}+{{{{(}^{n}}{{C}_{1}})}^{2}}+{{{{(}^{n}}{{C}_{2}})}^{2}}+..... $ $ +{{{{(}^{n}}{{C}_{5}})}^{2}} $ is equal to
Updated On: Jun 8, 2024
- 250
- 254
- 245
- 252
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The Correct Option is D
Solution and Explanation
$ {{{{(}^{n}}{{C}_{0}})}^{2}}+{{{{(}^{n}}{{C}_{1}})}^{2}}+{{{{(}^{n}}{{C}_{2}})}^{2}}+.....+{{{{(}^{n}}{{C}_{5}})}^{2}} $
$={{{{(}^{5}}{{C}_{0}})}^{2}}+{{{{(}^{5}}{{C}_{1}})}^{2}}+{{{{(}^{5}}{{C}_{2}})}^{2}}+{{{{(}^{n}}{{C}_{3}})}^{3}}+{{{{(}^{5}}{{C}_{4}})}^{4}} $ $ +{{{{(}^{5}}{{C}_{5}})}^{2}} $
$=1+25+100+100+25+1 $
$=252 $
$={{{{(}^{5}}{{C}_{0}})}^{2}}+{{{{(}^{5}}{{C}_{1}})}^{2}}+{{{{(}^{5}}{{C}_{2}})}^{2}}+{{{{(}^{n}}{{C}_{3}})}^{3}}+{{{{(}^{5}}{{C}_{4}})}^{4}} $ $ +{{{{(}^{5}}{{C}_{5}})}^{2}} $
$=1+25+100+100+25+1 $
$=252 $
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
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