Find the coefficient of
`x^2 y^3 z^4` in the expansion of `(ax -by +cz)^9`
(b) `a^2 b^3 c^4` d in the expansion of `(a -b-c +d)^10`

Answer

To solve the given problem, we need to find the coefficients of specific terms in the expansions of two different expressions. Let's break it down step by step.

### Part (a): Coefficient of \(x^2 y^3 z^4\) in the expansion of \((ax - by + cz)^9\)

1. **Identify the terms**: We need to find the coefficient of \(x^2 y^3 z^4\) in the expansion of \((ax - by + cz)^9\). Here, we can see that:
- \(x\) corresponds to \(ax\)
- \(y\) corresponds to \(-by\)
- \(z\) corresponds to \(cz\)

2. **Use the multinomial theorem**: The general term in the expansion of \((a_1 + a_2 + a_3)^n\) is given by:
\[ \frac{n!}{k_1! k_2! k_3!} a_1^{k_1} a_2^{k_2} a_3^{k_3} \]
where \(k_1 + k_2 + k_3 = n\).

3. **Assign values**: In our case, we want:
- \(k_1 = 2\) (for \(x^2\))
- \(k_2 = 3\) (for \(y^3\))
- \(k_3 = 4\) (for \(z^4\))
- \(n = 9\)

4. **Calculate the multinomial coefficient**:
\[ \text{Coefficient} = \frac{9!}{2!3!4!} \]

5. **Calculate factorials**:
- \(9! = 362880\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)

6. **Substitute and simplify**:
\[ \text{Coefficient} = \frac{362880}{2 \times 6 \times 24} = \frac{362880}{288} = 1260 \]

7. **Include the variables**: The term corresponding to \(x^2 y^3 z^4\) is:
\[ (a^2)(-b)^3(c)^4 = a^2(-b)^3c^4 = -a^2b^3c^4 \]

8. **Final coefficient**:
\[ \text{Final Coefficient} = 1260 \times (-1) = -1260 \]

### Part (b): Coefficient of \(a^2 b^3 c^4 d\) in the expansion of \((a - b - c + d)^{10}\)

1. **Identify the terms**: We need to find the coefficient of \(a^2 b^3 c^4 d\) in the expansion of \((a - b - c + d)^{10}\).

2. **Assign values**: Here, we want:
- \(k_1 = 2\) (for \(a^2\))
- \(k_2 = 3\) (for \(-b^3\))
- \(k_3 = 4\) (for \(-c^4\))
- \(k_4 = 1\) (for \(d\))
- \(n = 10\)

3. **Calculate the multinomial coefficient**:
\[ \text{Coefficient} = \frac{10!}{2!3!4!1!} \]

4. **Calculate factorials**:
- \(10! = 3628800\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)
- \(1! = 1\)

5. **Substitute and simplify**:
\[ \text{Coefficient} = \frac{3628800}{2 \times 6 \times 24 \times 1} = \frac{3628800}{288} = 12600 \]

6. **Include the signs**: The term corresponding to \(a^2 b^3 c^4 d\) is:
\[ (a^2)(-b)^3(-c)^4(d) = a^2(-1)^3b^3(-1)^4c^4d = -a^2b^3c^4d \]

7. **Final coefficient**:
\[ \text{Final Coefficient} = 12600 \times (-1) = -12600 \]

### Summary of Results:
- Coefficient of \(x^2 y^3 z^4\) in \((ax - by + cz)^9\) is \(-1260\).
- Coefficient of \(a^2 b^3 c^4 d\) in \((a - b - c + d)^{10}\) is \(-12600\).