HELP NEEDED !!

Can anyone please help me in solving this ∑i=020∑j=i+120(20i)(20j)\displaystyle{\sum _{ i=0 }^{ 20 }{ \sum _{ j=i+1 }^{ 20 }{ { { \left( \begin{matrix} 20 \\ i \end{matrix} \right) } } } } \left( \begin{matrix} 20 \\ j \end{matrix} \right) }

Its answer is 240−(4020)2\frac { { 2 }^{ 40 }-\left( \begin{matrix} 40 \\ 20 \end{matrix} \right) }{ 2 }

#Combinatorics #BinomialTheorem

Note by Vighnesh Raut
6 years, 2 months ago

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1 vote

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Comments

∑i=020∑j=i+120(20i)(20j)\displaystyle{\sum _{ i=0 }^{ 20 }{ \sum _{ j=i+1 }^{ 20 }{ { { \left( \begin{matrix} 20 \\ i \end{matrix} \right) } } } } \left( \begin{matrix} 20 \\ j \end{matrix} \right) } =∑i=020(20i)×[(20i+1)+(20i+2)+.....+(2020)]= \displaystyle \sum_{i=0}^{20} \binom{20}{i} \times \left[ \binom{20}{i+1}+\binom{20}{i+2}+.....+\binom{20}{20}\right] = Sum of the product of every possible combinations of two out of(200),(201),....,(2020)=λ(say)=\text{ Sum of the product of every possible combinations of two out of} \binom{20}{0} , \binom{20}{1}, ...., \binom{20}{20}=\lambda (say)

[(200)+(201)+....+(2020)]2=(200)2+(201)2+....+(2020)2+2λ\left[ \binom{20}{0} +\binom{20}{1}+ ....+ \binom{20}{20}\right]^2= \binom{20}{0}^2+ \binom{20}{1}^2+ ....+ \binom{20}{20}^2 +2 \lambda

240=(4020)+2λ2^{40}=\binom{40}{20}+2\lambda

  ⟹  λ=240−(4020)2\implies \lambda=\dfrac{2^{40}-\binom{40}{20}}{2}

Note :

∑k=0n(nk)2=(2nn)\displaystyle \sum_{k=0}^n \binom{n}{k}^2=\binom{2n}{n}

@Vighnesh Raut

Sandeep Bhardwaj - 6 years, 2 months ago

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Thank you so much sir.... It is a very detailed solution...Understood the process..Once again thanks..

Vighnesh Raut - 6 years, 2 months ago

Thanks Sir for the detailed solution.

Harshvardhan Mehta - 6 years, 2 months ago

From where did you get this question?

Adarsh Kumar - 6 years, 2 months ago

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It came in my mock mains test..

Vighnesh Raut - 6 years, 2 months ago

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Ok,which coaching centre?

Adarsh Kumar - 6 years, 2 months ago

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@Adarsh Kumar – Career Launcher..

Vighnesh Raut - 6 years, 2 months ago

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@Vighnesh Raut – ok thanx!

Adarsh Kumar - 6 years, 2 months ago

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@Adarsh Kumar – where have you joined??

Vighnesh Raut - 6 years, 2 months ago

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@Vighnesh Raut – I am 14 right now(going to be 15 on may 12),I don't go to any coaching centre.Sorry.

Adarsh Kumar - 6 years, 2 months ago

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@Adarsh Kumar – oh ok....

Vighnesh Raut - 6 years, 2 months ago
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