Question

If in the expansion of (1 + x)ⁿ, the coefficients of pth and qth terms are equal, prove that p + q = n + 2, where  \[p \neq q\]

 

Answer 1

\[\text{ Coefficients of the pth and qth terms are }  ^{n}{}{C}_{p - 1} \text{ and }^{n}{}{C}_{q - 1} \text{ respectively } . \]
\[\text{ Thus, we have: } \]
\[ ^{n}{}{C}_{p - 1} =^{n}{}{C}_{q - 1} \]
\[ \Rightarrow p - 1 = q - 1 \text{ or,}  p - 1 + q - 1 = n [ \because ^{n}{}{C}_r =^n C_s \Rightarrow r = s \text{ or, }  r + s = n]\]
\[ \Rightarrow p = q\text{ or } , p + q = n + 2\] 

If \[p \neq q\], then \[p + q = n + 2\]