Question
The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 + +100(1+x)^ 100 is equal to
JEE Main
Mathematics
Binomial Theorem
2026
JEE Main 2026 (22 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 100 ^ 100 C _ 49 - ^ 100 C _ 48
- B. ^ 100 C _ 50 + ^ 101 C _ 49
- C. ^ 100 ^ 101 C _ 49 - ^ 101 C _ 50
- D. 100 ^ 100 C _ 49 - ^ 100 C _ 50
Answer
C. ^ 100 ^ 101 C _ 49 - ^ 101 C _ 50
Step-by-step solution
Let 1 + x = r S = 1 r + 2 r^2 + 3 r^3 + + 100r^ 100 ......(1) (AGP) rS = 1 r^2 + 2 r^3 + + 99r^ 100 + 100r^ 101 ......(2) (1) - (2) gives S = - (1+x)^ 101 x^2 + 1 x^2 + 100(1+x)^ 101 x coefficient x^ 48 in S = - coefficient of x^ 48 in (1+x)^ 101 x^2 + 100 Coefficient of x^ 48 in (1+x)^ 101 x = 100 ^ 101 C_ 49 - ^ 101 C_ 50
Related: Mathematics — Binomial Theorem · All PYQ Banks