Question
Let S = 1 25! + 1 3!23! + 1 5!21! + up to 13 terms. If 13 ~S = 2^ k n! , k ~N , then n+k is equal to
JEE Main
Mathematics
Binomial Theorem
2026
JEE Main 2026 (24 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 50
- B. 49
- C. 52
- D. 51
Answer
B. 49
Step-by-step solution
The general term is 1 (2r-1)!(27-2r)! for r=1 to 13. Multiply and divide by 26!: 1 (2r-1)!(27-2r)! = 1 26! 26 2r-1 S = 1 26! _ r=1 ^ 13 26 2r-1 = 1 26! [ 26 1 + 26 3 + + 26 25 ] Sum of odd-indexed binomial coefficients: 26 odd = 2^ 25 S = 2^ 25 26! 13S = 13 2^ 25 26! = 13 2^ 25 26 25! = 2^ 25 2 25! = 2^ 24 25! So k = 24,\; n = 25, giving n + k = 49.
Related: Mathematics — Binomial Theorem · All PYQ Banks