State whether this statement is true or not: The number of term in the expansion of `(a + b)^(n)`, where `n in N`, is one less than the power `n`

Answer

To determine whether the statement "The number of terms in the expansion of \( (a + b)^n \), where \( n \in \mathbb{N} \), is one less than the power \( n \)" is true or false, we will analyze the binomial expansion step by step.

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The binomial theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
where \( \binom{n}{k} \) is the binomial coefficient.

2. **Identifying the Range of k**:
In the expansion, \( k \) takes values from \( 0 \) to \( n \). This means that \( k \) can take on \( n + 1 \) different values (including both endpoints).

3. **Counting the Terms**:
Each unique value of \( k \) corresponds to a unique term in the expansion. Therefore, the total number of terms in the expansion of \( (a + b)^n \) is \( n + 1 \).

4. **Analyzing the Given Statement**:
The statement claims that the number of terms is one less than the power \( n \). Mathematically, this would mean:
\[ \text{Number of terms} = n - 1 \]
However, we found that:
\[ \text{Number of terms} = n + 1 \]

5. **Conclusion**:
Since \( n + 1 \) is not equal to \( n - 1 \), the statement is false. The correct assertion is that the number of terms in the expansion of \( (a + b)^n \) is one more than the power \( n \).

### Final Answer:
The statement is **false**. The number of terms in the expansion of \( (a + b)^n \) is \( n + 1 \), which is one more than the power \( n \).