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

Its answer is \[\frac { { 2 }^{ 40 }-\left( \begin{matrix} 40 \\ 20 \end{matrix} \right) }{ 2 } \]

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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

\[\left[ \binom{20}{0} +\binom{20}{1}+ ....+ \binom{20}{20}\right]^2= \binom{20}{0}^2+ \binom{20}{1}^2+ ....+ \binom{20}{20}^2 +2 \lambda\]

\[2^{40}=\binom{40}{20}+2\lambda\]

\[\implies \lambda=\dfrac{2^{40}-\binom{40}{20}}{2}\]

Note :

\[\displaystyle \sum_{k=0}^n \binom{n}{k}^2=\binom{2n}{n}\]

@Vighnesh Raut

Log in to reply

Thanks Sir for the detailed solution.

Log in to reply

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

Log in to reply

From where did you get this question?

Log in to reply

It came in my mock mains test..

Log in to reply

Ok,which coaching centre?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply