1
JEE Main 2022 (Online) 28th June Morning Shift
+4
-1
Out of Syllabus

If

$$\sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {{}^{30}{C_k}} \right)\left( {{}^{30}{C_{k - 1}}} \right) = {{\alpha (60!)} \over {(30!)(31!)}}} }$$,

where $$\alpha$$ $$\in$$ R, then the value of 16$$\alpha$$ is equal to

A
1411
B
1320
C
1615
D
1855
2
JEE Main 2022 (Online) 26th June Morning Shift
+4
-1

The remainder when (2021)2023 is divided by 7 is :

A
1
B
2
C
5
D
6
3
JEE Main 2022 (Online) 25th June Evening Shift
+4
-1

The coefficient of x101 in the expression $${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}$$, x > 0, is

A
501C101 (5)399
B
501C101 (5)400
C
501C100 (5)400
D
500C101 (5)399
4
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1

If $${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}$$, then the remainder when K is divided by 6 is :

A
1
B
2
C
3
D
5
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