The coefficient of `x^2 y^5 z^3` in the expansion of `(2x + y + 3z)^10` is

Options

A. `(10!)/(2!3!5!)`
B. `(10!)/(2!3!5!)xx2^(2) xx3^(3)`
C. `(10!)/(2!3!5!)xx2^(3) xx3^(2)`
D. `10! xx2^(2) xx3^(3)`

Explanation

To find the coefficient of \(x^2 y^5 z^3\) in the expansion of \((2x + y + 3z)^{10}\), we can use the multinomial expansion formula. Here’s a step-by-step solution:

### Step 1: Identify the terms
The expression we are expanding is \((2x + y + 3z)^{10}\). We need to find the coefficient of the term \(x^2 y^5 z^3\).

### Step 2: Set up the multinomial expansion
In the multinomial expansion, the general term can be expressed as:
\[ \frac{n!}{r_1! r_2! r_3!} (a_1)^{r_1} (a_2)^{r_2} (a_3)^{r_3} \]
where \(n = r_1 + r_2 + r_3\).

In our case:
- \(n = 10\)
- \(a_1 = 2x\), \(a_2 = y\), \(a_3 = 3z\)
- We need \(r_1 = 2\) (for \(x^2\)), \(r_2 = 5\) (for \(y^5\)), and \(r_3 = 3\) (for \(z^3\)).

### Step 3: Calculate \(r_1 + r_2 + r_3\)
Check that:
\[ r_1 + r_2 + r_3 = 2 + 5 + 3 = 10 \]
This confirms that we have the correct values for \(r_1\), \(r_2\), and \(r_3\).

### Step 4: Write the general term
The general term for our expansion is:
\[ \frac{10!}{2!5!3!} (2x)^2 (y)^5 (3z)^3 \]

### Step 5: Substitute the values
Now substitute \(r_1\), \(r_2\), and \(r_3\) into the general term:
\[ = \frac{10!}{2!5!3!} (2^2 x^2) (y^5) (3^3 z^3) \]

### Step 6: Simplify the expression
Calculating the powers:
\[ = \frac{10!}{2!5!3!} \cdot 4 \cdot x^2 \cdot y^5 \cdot 27 \cdot z^3 \]
Combine the constants:
\[ = \frac{10!}{2!5!3!} \cdot 108 \cdot x^2 \cdot y^5 \cdot z^3 \]

### Step 7: Calculate the coefficient
Now we need to compute the coefficient:
\[ \text{Coefficient} = \frac{10!}{2!5!3!} \cdot 108 \]

### Step 8: Calculate factorials
Calculating the factorials:
- \(10! = 3628800\)
- \(2! = 2\)
- \(5! = 120\)
- \(3! = 6\)

Now substitute these values:
\[ = \frac{3628800}{2 \cdot 120 \cdot 6} \cdot 108 \]

### Step 9: Simplify further
Calculating the denominator:
\[ = 2 \cdot 120 \cdot 6 = 1440 \]
Now divide:
\[ = \frac{3628800}{1440} \cdot 108 = 2520 \cdot 108 \]

### Step 10: Final multiplication
Calculating the final multiplication:
\[ = 272160 \]

Thus, the coefficient of \(x^2 y^5 z^3\) in the expansion of \((2x + y + 3z)^{10}\) is **272160**.

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