The number of terms in the expansion of `(a + b + c)^25` is -

Options

A. 26
B. 51
C. 251
D. 351

Answer

To find the number of terms in the expansion of \((a + b + c)^{25}\), 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 power of the expression and \(r\) is the number of different variables in the expression.

### Step-by-Step Solution:

1. **Identify the variables and power**:
- In the expression \((a + b + c)^{25}\), we have:
- \(n = 25\) (the power)
- \(r = 3\) (the number of terms: \(a\), \(b\), and \(c\))

2. **Apply the formula**:
- Substitute \(n\) and \(r\) into the formula:
\[ \text{Number of terms} = \binom{25 + 3 - 1}{3 - 1} = \binom{27}{2} \]

3. **Calculate \(\binom{27}{2}\)**:
- The binomial coefficient \(\binom{27}{2}\) is calculated as follows:
\[ \binom{27}{2} = \frac{27!}{2!(27 - 2)!} = \frac{27!}{2! \cdot 25!} \]
- This simplifies to:
\[ \binom{27}{2} = \frac{27 \times 26}{2 \times 1} = \frac{702}{2} = 351 \]

4. **Conclusion**:
- Therefore, the number of terms in the expansion of \((a + b + c)^{25}\) is \(351\).

### Final Answer:
The number of terms in the expansion of \((a + b + c)^{25}\) is \(351\).