$\sum_{\substack{i, j=0 \\ i \neq j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j}$ is equal to
$\sum_{\substack{i, j=0 \\ i \neq j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j}$ is equal to
- [JEE MAIN 2022]
- A
$2^{2 n }-{ }^{2 n } C _{ n }$
- B
$2^{2 n -1}-^{2 n -1} C _{ n -1}$
- C
$2^{2 n }-\frac{1}{2}{ }^{2 n } C _{ n }$
- D
$2^{ n -1}+{ }^{2 n -1} C _{ n }$
Similar Questions
$\sum\limits_{k = 0}^{10} {^{20}{C_k} = } $
$\sum\limits_{k = 0}^{10} {^{20}{C_k} = } $
The sum of the coefficients of even power of $x$ in the expansion of ${(1 + x + {x^2} + {x^3})^5}$ is
The sum of the coefficients of even power of $x$ in the expansion of ${(1 + x + {x^2} + {x^3})^5}$ is
If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ... + {C_n}{x^n}$, then the value of ${C_0} + {C_2} + {C_4} + {C_6} + .....$ is
If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ... + {C_n}{x^n}$, then the value of ${C_0} + {C_2} + {C_4} + {C_6} + .....$ is
Total number of terms in the expansion of $\left[ {{{\left( {1 + x} \right)}^{100}} + {{\left( {1 + {x^2}} \right)}^{100}}{{\left( {1 + {x^3}} \right)}^{100}}} \right]$ is
Total number of terms in the expansion of $\left[ {{{\left( {1 + x} \right)}^{100}} + {{\left( {1 + {x^2}} \right)}^{100}}{{\left( {1 + {x^3}} \right)}^{100}}} \right]$ is
The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+$ $x^4(1+x)^{96}+\ldots \ldots . .+x^{54}(1+x)^{46}$ is ${ }^{99} \mathrm{C}_p-{ }^{46} \mathrm{C}_{\mathrm{q}}$.
Then a possible value to $\mathrm{p}+\mathrm{q}$ is :
The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+$ $x^4(1+x)^{96}+\ldots \ldots . .+x^{54}(1+x)^{46}$ is ${ }^{99} \mathrm{C}_p-{ }^{46} \mathrm{C}_{\mathrm{q}}$.
Then a possible value to $\mathrm{p}+\mathrm{q}$ is :
- [JEE MAIN 2024]