what are the coefficients for the binomial expansion of (a + b)^3?\n1, 4, 6, 4, 1\n1, 1\n1, 2, 1\n1, 3, 3, 1

Explanation:

Step1: Recall binomial theorem

The binomial theorem states that $(a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}$, where $\binom{n}{k}=\frac{n!}{k!(n - k)!}$. For $(a + b)^3$, $n = 3$.

Step2: Calculate coefficients for $k = 0$

When $k = 0$, $\binom{3}{0}=\frac{3!}{0!(3 - 0)!}=\frac{3!}{3!}=1$.

Step3: Calculate coefficients for $k = 1$

When $k = 1$, $\binom{3}{1}=\frac{3!}{1!(3 - 1)!}=\frac{3!}{1!2!}=\frac{3\times2!}{2!}=3$.

Step4: Calculate coefficients for $k = 2$

When $k = 2$, $\binom{3}{2}=\frac{3!}{2!(3 - 2)!}=\frac{3!}{2!1!}=\frac{3\times2!}{2!}=3$.

Step5: Calculate coefficients for $k = 3$

When $k = 3$, $\binom{3}{3}=\frac{3!}{3!(3 - 3)!}=\frac{3!}{3!0!}=1$.

Answer:

D. 1, 3, 3, 1