The number of terms in the expansion if `(a+b+c)^(12)` is

Options

A. 90
B. 91
C. 81
D. 80

Answer

To find the number of terms in the expansion of \((a+b+c)^{12}\), we can use the formula for the number of terms in the expansion of \((x_1 + x_2 + ... + x_r)^n\), which is given by:

\[ \text{Number of terms} = \binom{n + r - 1}{r - 1} \]

where \(n\) is the exponent and \(r\) is the number of different variables (terms).

### Step 1: Identify \(n\) and \(r\)
In our case:
- The exponent \(n = 12\)
- The number of different variables \(r = 3\) (which are \(a\), \(b\), and \(c\))

### Step 2: Substitute into the formula
Now, we substitute \(n\) and \(r\) into the formula:

\[ \text{Number of terms} = \binom{12 + 3 - 1}{3 - 1} = \binom{12 + 2}{2} = \binom{14}{2} \]

### Step 3: Calculate \(\binom{14}{2}\)
Now we calculate \(\binom{14}{2}\):

\[ \binom{14}{2} = \frac{14 \times 13}{2 \times 1} = \frac{182}{2} = 91 \]

### Conclusion
Thus, the number of terms in the expansion of \((a+b+c)^{12}\) is **91**.

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