Question
The value of ^ 100 C _ 50 51 + ^ 100 C _ 51 52 + .+ ^ 100 C _ 100 101 is :
JEE Main
Mathematics
Binomial Theorem
2026
JEE Main 2026 (23 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 2^ 101 100
- B. 2^ 100 100
- C. 2^ 101 101
- D. 2^ 100 101
Answer
D. 2^ 100 101
Step-by-step solution
Using n r r+1 = 1 n+1 n+1 r+1 : _ r=50 ^ 100 100 r r+1 = 1 101 _ r=50 ^ 100 101 r+1 = 1 101 _ k=51 ^ 101 101 k . Since 101 is odd, by symmetry _ k=51 ^ 101 101 k = _ k=0 ^ 50 101 k = 2^ 101 2 = 2^ 100 . Required value = 2^ 100 101 .
Related: Mathematics — Binomial Theorem · All PYQ Banks