Binomial Theorem & Pascal's Triangle

Use the binomial theorem to find the term that will give \(x^{10}\) in the expansion of \(\left(\frac{1}{3}x^4 - \frac{1}{2x^2}\right)^7\).

We only want to find the coefficient of the term in \(x^{10}\) so we don't need the complete expansion. Let us examine the indices of the \(x\)-variable.

If we have \((x^4)^6\left(\frac{1}{x^2}\right)\) we will get \(x^{22}\). To get \(x^{10}\), we have to raise \(x^4\) to the 4th power, which means we have to raise \(\frac{1}{x^2}\) to the 3rd power since the indices must sum up to 7.

So, we let \(a = \frac{1}{3}x^4\), \(b = -\frac{1}{2x^2}\), \(n = 7\), \(n - k = 4\) and \(k = 3\).

Substitute \(a\), \(b\), \(n\), \(k\), \(n - k\) into \(\binom{n}{k}a^{n-k}b^{k}\) to get the term with \(x^{10}\):

Evaluate \(\binom{7}{3}\), \(\left(\frac{1}{3}x^{4}\right)^{4}\) and \(\left(-\frac{1}{2x^{2}}\right)^{3}\):

Evaluate the product: