Question

The binomial coefficient of middle term in the expansion of $\left(\frac{2}{x}-\frac{x}{2}\right)^9$, can be

• A. 108
• B. 126
• C. 132
• D. 164

Answer 1

Answer: (B)

126

Answer 2

The expansion of $\left(\frac{2}{x}-\frac{x}{2}\right)^9$ has $n+1=10$ terms. For an even number of terms, there are two middle terms: the 5th and 6th. The general term is $T_{r+1} = \binom{n}{r} a^{n-r} b^r$. For the 5th term, $r=4$, giving coefficient $\binom{9}{4}$. For the 6th term, $r=5$, giving coefficient $\binom{9}{5}$. Since $\binom{9}{4} = \binom{9}{5}$, the binomial coefficient is unique. Calculating $\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126$.