Question

question
find the coefficient of $x^{2}y^{6}$ in the expansion of $(3x - y)^{8}$.
select the correct answer below:
63
252
1485
257
-246

Explanation:

Step1: Recall binomial theorem

The binomial - theorem states that \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\). In the expansion of \((3x-y)^8\), we have \(a = 3x\), \(b=-y\), and \(n = 8\). We want to find the coefficient of \(x^{2}y^{6}\), so we set \(n - k=2\) and \(k = 6\).

Step2: Calculate the binomial coefficient

The binomial coefficient \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\). Here, \(n = 8\) and \(k = 6\), so \(\binom{8}{6}=\frac{8!}{6!(8 - 6)!}=\frac{8!}{6!2!}=\frac{8\times7}{2\times1}=28\).

Step3: Calculate the coefficient of the term

The term in the expansion of \((3x-y)^8\) is \(\binom{8}{6}(3x)^{8 - 6}(-y)^{6}\). Substitute \(\binom{8}{6}=28\), \((3x)^{2}=9x^{2}\), and \((-y)^{6}=y^{6}\) into the term. We get \(28\times9x^{2}y^{6}=252x^{2}y^{6}\).

Answer:

252