The coefficient of `x^2y^3z^4` in `(2x + y - z)^9` is

Answer

To find the coefficient of \( x^2y^3z^4 \) in the expansion of \( (2x + y - z)^9 \), we can use the multinomial expansion formula. Here’s a step-by-step solution:

### Step 1: Identify the terms in the expansion
The expression we are expanding is \( (2x + y - z)^9 \). We need to find the coefficient of the term \( x^2y^3z^4 \).

### Step 2: Set up the multinomial expansion
The general term in the expansion of \( (a + b + c)^n \) is given by:
\[ T = \frac{n!}{r_1! r_2! r_3!} a^{r_1} b^{r_2} c^{r_3} \]
where \( r_1 + r_2 + r_3 = n \).

In our case:
- \( a = 2x \)
- \( b = y \)
- \( c = -z \)
- \( n = 9 \)

### Step 3: Determine \( r_1, r_2, r_3 \)
We need:
- \( r_1 \) (the exponent of \( 2x \)) to be 2 (for \( x^2 \))
- \( r_2 \) (the exponent of \( y \)) to be 3 (for \( y^3 \))
- \( r_3 \) (the exponent of \( -z \)) to be 4 (for \( z^4 \))

Thus, we have:
- \( r_1 = 2 \)
- \( r_2 = 3 \)
- \( r_3 = 4 \)

### Step 4: Verify the sum of the exponents
Check that \( r_1 + r_2 + r_3 = 2 + 3 + 4 = 9 \), which is equal to \( n \).

### Step 5: Substitute into the multinomial formula
Now we can substitute these values into the multinomial formula:
\[ T = \frac{9!}{2!3!4!} (2x)^2 (y)^3 (-z)^4 \]

### Step 6: Calculate the terms
Calculating each part:
- \( (2x)^2 = 4x^2 \)
- \( (y)^3 = y^3 \)
- \( (-z)^4 = z^4 \)

Now substituting back:
\[ T = \frac{9!}{2!3!4!} \cdot 4x^2 \cdot y^3 \cdot z^4 \]

### Step 7: Calculate the coefficient
The coefficient is:
\[ \text{Coefficient} = \frac{9!}{2!3!4!} \cdot 4 \]

### Step 8: Calculate factorials
Now, we calculate:
- \( 9! = 362880 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 4! = 24 \)

So:
\[ 2!3!4! = 2 \cdot 6 \cdot 24 = 288 \]

### Step 9: Final calculation
Now we can find the coefficient:
\[ \text{Coefficient} = \frac{362880}{288} \cdot 4 \]
Calculating \( \frac{362880}{288} = 1260 \):
\[ \text{Coefficient} = 1260 \cdot 4 = 5040 \]

### Final Answer
The coefficient of \( x^2y^3z^4 \) in the expansion of \( (2x + y - z)^9 \) is **5040**.