Question

If the constant term, in binomial expansion of \(\left(2x^r + \frac{1}{x^2}\right)^{10}\) is 180, then \(r\) is equal to __________________.

Hint:

Write the general term, set the power of \(x\) to 0 to relate \(k\) and \(r\), and match the coefficient with 180.

Answer 1

Correct Answer: 8

General term: \(T_{k+1} = \binom{10}{k}(2x^r)^{10-k}\left(\frac{1}{x^2}\right)^k = \binom{10}{k}2^{10-k}x^{r(10-k) - 2k}\).
Constant term: \(r(10 - k) = 2k\), so \(r = \frac{2k}{10 - k}\).
The constant term is \(\binom{10}{k}2^{10-k} = 180\).
Try \(k = 8\): \(\binom{10}{8}2^2 = 45 \times 4 = 180\) and \(r = \frac{16}{2} = 8\).
Answer: 8