Σ_{mathrm{k}=0}^{20}left({ }^{20} mathrm{C}_{mathrm{k}}right)^{2} is equal to :

English

Gujarati

Hindi

7.Binomial Theorem

hard

$\sum_{\mathrm{k}=0}^{20}\left({ }^{20} \mathrm{C}_{\mathrm{k}}\right)^{2}$ is equal to :

A

${ }^{40} \mathrm{C}_{21}$

B

${ }^{40} \mathrm{C}_{19}$

C

${ }^{40} \mathrm{C}_{20}$

D

${ }^{41} \mathrm{C}_{20}$

(JEE MAIN-2021)

Solution

$\sum_{\mathrm{k}=0}^{20}{ }^{20} \mathrm{C}_{\mathrm{k}} \cdot{ }^{20} \mathrm{C}_{20-\mathrm{k}}$

sum of suffix is const. so summation will be ${ }^{40} \mathrm{C}_{20}$

Standard 11

Mathematics

Share

Similar Questions

Let $\alpha=\sum_{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)$ and $\beta=\sum_{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)$. If $5 \alpha=6 \beta$, then $n$ equals

normal

(JEE MAIN-2024)

The coefficient of $x^{49}$ in the expansion of $(x – 1)$$\left( {x\, – \,\frac{1}{2}\,} \right)$$\left( {x\, – \,\frac{1}{{{2^2}}}\,} \right)$ …..$\left( {x\, – \,\frac{1}{{{2^{49}}}}\,} \right)$ is equal to

normal

If $\sum_{ r =0}^5 \frac{{ }^{11} C _{2 r +1}}{2 r +2}=\frac{ m }{ n }, \operatorname{gcd}( m , n )=1$, then $m – n$ is equal to _____

hard

(JEE MAIN-2025)

If $\sum_{r=1}^{10} r !\left( r ^{3}+6 r ^{2}+2 r +5\right)=\alpha(11 !),$ then the value of $\alpha$ is equal to …… .

hard

(JEE MAIN-2021)

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+ x )^{ n +2}$, which are in the ratio $1: 3: 5$, is equal to

hard

(JEE MAIN-2023)