Let's focus on solving the math question you uploaded:

Part (a)

We are asked to find the coefficient of $x^6 y^2$ in the expansion of $(3x^2 + y)^5$ using the binomial theorem.

The binomial theorem states that for any expansion of the form $(a + b)^n$, we have:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

In this case:

• $a = 3x^2$
• $b = y$
• $n = 5$

So, the general term in the expansion is given by:

$\binom{5}{k} (3x^2)^{5-k} y^k$

Simplifying this, we get:

$\binom{5}{k} 3^{5-k} x^{2(5-k)} y^k$

We need to find the term where $x^6 y^2$ appears. This means:

• The power of $x$ is 6, so $2(5-k) = 6$, which gives $k = 2$.
• The power of $y$ is 2, which matches $k = 2$.

Now, substitute $k = 2$ into the general term:

$\binom{5}{2} 3^{5-2} x^{2(5-2)} y^2 = \binom{5}{2} 3^3 x^6 y^2$

Now calculate:

$\binom{5}{2} = \frac{5!}{2!(5-2)!} = 10, \quad 3^3 = 27$

So, the coefficient of $x^6 y^2$ is:

$10 \times 27 = 270$

Thus, the coefficient of $x^6 y^2$ in the expansion of $(3x^2 + y)^5$ is 270.

Part (b)

We are asked to interpret the expression:

$[x_1 x_2 x_3^2] (x_1 + 2x_2 + 3x_3)^4$

This refers to a multinomial expansion. The multinomial expansion of $(x_1 + 2x_2 + 3x_3)^4$ can be written as:

$(x_1 + 2x_2 + 3x_3)^4 = \sum_{i_1 + i_2 + i_3 = 4} \binom{4}{i_1, i_2, i_3} (x_1)^{i_1} (2x_2)^{i_2} (3x_3)^{i_3}$

The notation $[x_1 x_2 x_3^2]$ means we are looking for the term where $x_1^1 x_2^1 x_3^2$ appears. This corresponds to:

• $i_1 = 1$, $i_2 = 1$, and $i_3 = 2$.

For these values, the multinomial coefficient is:

$\binom{4}{1, 1, 2} = \frac{4!}{1! 1! 2!} = 12$

Now, the corresponding term is:

$12 \cdot (x_1)^1 \cdot (2x_2)^1 \cdot (3x_3)^2 = 12 \cdot 2 \cdot 9 \cdot x_1 x_2 x_3^2$

Thus, the coefficient of $x_1 x_2 x_3^2$ is:

$12 \times 2 \times 9 = 216$

The interpretation of this expression is that it represents the number of ways to select certain objects, accounting for different "weights" or counts of selections from each group, similar to a combinatorial counting problem.

Would you like further details or explanations? Here are 5 related questions that expand on this information:

1. How does the multinomial theorem generalize the binomial theorem?
2. Can the binomial coefficient $\binom{n}{k}$ be used for other types of expansions?
3. How do we determine the total number of terms in a multinomial expansion?
4. What other types of problems can be solved using multinomial expansions?
5. How does the combinatorial interpretation of the binomial and multinomial theorems connect to probability theory?

Tip: Always check the powers of variables carefully when applying the binomial theorem to avoid mismatches.