Number of terms in (a + b)ⁿ expansion is:

To find the number of terms in the expansion of $(a + b)^n$, we use the **Binomial Theorem**. The Binomial Theorem states that for any positive integer $n$: $$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$$ Expanding this sum, we get: $$(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \dots + \binom{n}{n} a^0 b^n$$ By looking at the index $r$ of the binomial coefficients $\binom{n}{r}$, we can see that: 1. The first term corresponds to $r = 0$. 2. The second term corresponds to $r = 1$. 3. The third term corresponds to $r = 2$. 4. This continues until the last term, which corresponds to $r = n$. To count the total number of terms, we count the integers in the set $\{0, 1, 2, \dots, n\}$. The number of elements is calculated as: $$\text{Total Terms} = (\text{End} - \text{Start}) + 1$$ $$\text{Total Terms} = (n - 0) + 1 = n + 1$$ Thus, the number of terms in the expansion of $(a + b)^n$ is $n + 1$.