The total number of terms in the expansion of
`(2x - y + 4z)^(12)`,is

Options

A. 90
B. 91
C. 13
D. none of these

Explanation

To find the total number of terms in the expansion of \((2x - y + 4z)^{12}\), we can follow these steps:

### Step 1: Identify the number of variables
In the expression \((2x - y + 4z)\), we have three variables: \(2x\), \(-y\), and \(4z\). Therefore, the number of variables \(r = 3\).

### Step 2: Identify the power of the expression
The power of the expression is \(n = 12\).

### Step 3: Use the formula for the number of terms in a multinomial expansion
The total number of terms in the expansion of \((a_1 + a_2 + a_3)^{n}\) is given by the formula:
\[ \text{Total number of terms} = \binom{n + r - 1}{r - 1} \]
where \(n\) is the power and \(r\) is the number of variables.

### Step 4: Substitute the values into the formula
Substituting \(n = 12\) and \(r = 3\) into the formula:
\[ \text{Total number of terms} = \binom{12 + 3 - 1}{3 - 1} = \binom{14}{2} \]

### Step 5: Calculate \(\binom{14}{2}\)
Now, we calculate \(\binom{14}{2}\):
\[ \binom{14}{2} = \frac{14!}{2!(14-2)!} = \frac{14 \times 13}{2 \times 1} = \frac{182}{2} = 91 \]

### Conclusion
Thus, the total number of terms in the expansion of \((2x - y + 4z)^{12}\) is **91**.

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