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 in the expression.

### Step-by-Step Solution:

1. **Identify \(n\) and \(r\)**:
- In our case, the expression is \((a + b + c)^{12}\).
- Here, \(n = 12\) (the exponent) and \(r = 3\) (the number of variables: \(a\), \(b\), and \(c\)).

2. **Apply the formula**:
- Substitute \(n\) and \(r\) into the formula:
\[ \text{Number of terms} = \binom{12 + 3 - 1}{3 - 1} = \binom{12 + 2}{2} = \binom{14}{2} \]

3. **Calculate \(\binom{14}{2}\)**:
- The binomial coefficient \(\binom{14}{2}\) is calculated as follows:
\[ \binom{14}{2} = \frac{14!}{2!(14 - 2)!} = \frac{14!}{2! \cdot 12!} \]
- Simplifying this:
\[ = \frac{14 \times 13 \times 12!}{2 \times 1 \times 12!} = \frac{14 \times 13}{2 \times 1} = \frac{182}{2} = 91 \]

4. **Conclusion**:
- Therefore, the number of terms in the expansion of \((a + b + c)^{12}\) is **91**.