The number of the terms which are not similar in the expansion of `(L+M+N)^(6)`

Options

A. 7
B. 42
C. 28
D. 21

Answer

To find the number of distinct terms in the expansion of \((L + M + N)^6\), we can use the formula for the number of distinct terms in the expansion of \((x_1 + x_2 + ... + x_m)^n\), which is given by:

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

where:
- \(n\) is the exponent,
- \(m\) is the number of different variables in the expression.

### Step-by-step Solution:

1. **Identify the values of \(n\) and \(m\)**:
- In our case, the expression is \((L + M + N)^6\).
- Here, \(n = 6\) (the exponent) and \(m = 3\) (the number of different terms: \(L\), \(M\), and \(N\)).

2. **Substitute the values into the formula**:
- We need to calculate \(\binom{n + m - 1}{m - 1}\).
- Substituting the values, we have:
\[ \binom{6 + 3 - 1}{3 - 1} = \binom{8}{2} \]

3. **Calculate \(\binom{8}{2}\)**:
- The binomial coefficient \(\binom{8}{2}\) is calculated as follows:
\[ \binom{8}{2} = \frac{8!}{2!(8 - 2)!} = \frac{8!}{2! \cdot 6!} \]
- Simplifying this, we get:
\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = \frac{56}{2} = 28 \]

4. **Conclusion**:
- Therefore, the number of distinct terms in the expansion of \((L + M + N)^6\) is \(28\).

### Final Answer:
The number of distinct terms in the expansion of \((L + M + N)^6\) is **28**.