Binomial Theorem
Binomial Theorem
nta_abhyas_2025
Grade 11
Question:
The coefficient of $x^{10}$ in the expansion of $(1+x)^{10} + (1+x)^{10} + (1+x)^{12} + \ldots + (1+x)^{20}$ is
$^{31}C_{10} - ^{21}C_{10}$
$^{31}C_{11} - ^{21}C_{10}$
$^{30}C_{10} - ^{21}C_{10}$
$^{31}C_{10} - ^{20}C_{10}$
Step-by-Step Solution
Key Concept: When binomial coefficients are equal, $^nC_r = ^nC_s$ implies $r+s = n$, which determines the value of $n$.
From $^nC_{10} = ^nC_{11}$, we get $n = 21$. The coefficient of $x^{10}$ in $(^nC_0 + ^nC_1 + ... + ^nC_n)(^nC_{11} + ^nC_{12} + ... + ^nC_n) = ^{21}C_{10} - ^{21}C_{11} = ^{21}C_{11}$.
Correct Answer: 21