Find the coefficient of \(x^{4}\) in the binomial expansion of \((1 - 2x)^{6}\).
Answer Details
We can use the binomial theorem to find the coefficient of \(x^{4}\) in the expansion of \((1 - 2x)^{6}\). The binomial theorem states that the \(n\)th power of a binomial \((a + b)^{n}\) can be expressed as the sum of the terms of the form \(a^{n-r}b^{r}\), where \(r\) ranges from 0 to \(n\).
In this case, the binomial is \((1 - 2x)\), so we can expand it using the binomial theorem as follows:
\[(1 - 2x)^{6} = \sum_{r=0}^{6} {6 \choose r} (1)^{6-r} (-2x)^{r}\]
where \({6 \choose r}\) is the binomial coefficient.
To find the coefficient of \(x^{4}\), we need to find the term where \(r\) is 2. So, we can substitute \(r = 2\) into the formula above to get:
\[{6 \choose 2} (1)^{4} (-2x)^{2} = 15 \times 4x^{2} = 60x^{2}\]
However, we are looking for the coefficient of \(x^{4}\), not \(x^{2}\). Since the exponent of \(x\) is 2 in the term we just calculated, we need to multiply it by another power of \(x^{2}\) to get \(x^{4}\). This means we need to find the term where \(r\) is 4.
Substituting \(r = 4\) into the formula above, we get:
\[{6 \choose 4} (1)^{2} (-2x)^{4} = 15 \times 16x^{4} = 240x^{4}\]
Therefore, the coefficient of \(x^{4}\) in the binomial expansion of \((1 - 2x)^{6}\) is 240.
Hence, the answer is 240.