1
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1
Out of Syllabus

If $$a_r$$ is the coefficient of $$x^{10-r}$$ in the Binomial expansion of $$(1 + x)^{10}$$, then $$\sum\limits_{r = 1}^{10} {{r^3}{{\left( {{{{a_r}} \over {{a_{r - 1}}}}} \right)}^2}}$$ is equal to

A
3025
B
4895
C
5445
D
1210
2
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1
Out of Syllabus

If $${({}^{30}{C_1})^2} + 2{({}^{30}{C_2})^2} + 3{({}^{30}{C_3})^2}\, + \,...\, + \,30{({}^{30}{C_{30}})^2} = {{\alpha 60!} \over {{{(30!)}^2}}}$$ then $$\alpha$$ is equal to :

A
30
B
10
C
15
D
60
3
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1
Out of Syllabus

The value of $$\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}}$$ is

A
$${}^{44}{C_{23}}$$
B
$${}^{45}{C_{23}}$$
C
$${}^{44}{C_{22}}$$
D
$${}^{45}{C_{24}}$$
4
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1
Out of Syllabus

$$\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)$$ is equal to

A
$$22 !-21 !$$
B
$$22 !-2(21 !)$$
C
$$21 !-2(20 !)$$
D
$$21 !-20$$ !
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