Question

The middle term in the expansion of \[ (10x + x^{10})^{10} \] {is:}

• A. \( 10C5 \)
• B. \( 10C6 \)
• C. \( 10C5 x^{10} \)
• D. \( 10C5 x^{10} \)

Hint:

The middle term in a binomial expansion corresponds to the term where the exponent of \( x \) is the average of the highest and lowest exponents.

Answer 1

The Correct Option is A

Step 1: The binomial expansion of \( (10x + x^{10})^{10} \) is given by: \[ (10x + x^{10})^{10} = \sum_{r=0}^{10} \binom{10}{r} (10x)^{10-r} (x^{10})^r. \] Step 2: The general term is: \[ \binom{10}{r} (10x)^{10-r} (x^{10})^r = \binom{10}{r} 10^{10-r} x^{10-r + 10r}. \] This simplifies to: \[ \binom{10}{r} 10^{10-r} x^{10 + 9r}. \] Step 3: The middle term corresponds to \( r = 5 \), as the series has 11 terms. Thus, the middle term is: \[ \binom{10}{5} 10^5 x^{10 + 9(5)} = \binom{10}{5} 10^5 x^{55}. \] Step 4: Hence, the middle term is \( 10C5 \).