Number of terms in the expansion of (1+x³+x⁴)^8

The number of terms in the expansion of \((1 + x^3 + x^4)^8\) can be determined by recognizing that this is a multinomial expansion.

### Steps:
1. The expression \((1 + x^3 + x^4)^8\) consists of three terms: \(1\), \(x^3\), and \(x^4\).
2. Each term in the expansion will be of the form:
   \[
   1^a \cdot (x^3)^b \cdot (x^4)^c
   \]
   where \(a + b + c = 8\), and \(a\), \(b\), and \(c\) are non-negative integers representing how many times each of the three terms appears in the expansion.

3. The number of distinct terms corresponds to the number of different combinations of powers of \(x\) generated by different values of \(b\) and \(c\) (since \(x^3\) and \(x^4\) introduce powers of \(x\)).

4. The number of distinct terms can be found by considering the number of non-negative integer solutions to \(a + b + c = 8\). This is a standard stars and bars problem, and the number of solutions is given by:
   \[
   \text{Number of terms} = \binom{8 + 3 - 1}{3 - 1} = \binom{10}{2} = \frac{10 \times 9}{2} = 45
   \]
   
Thus, the number of terms in the expansion of \((1 + x^3 + x^4)^8\) is **45**.

Would you like further details on multinomial expansions or combinatorics? Here are some related questions:

1. How do we derive the multinomial expansion formula?
2. What are stars and bars in combinatorics, and how does it apply here?
3. How do the powers of \(x\) vary in this expansion?
4. What is the general formula for the number of terms in a multinomial expansion?
5. Can we calculate the coefficient of a specific term in this expansion?

**Tip:** In multinomial expansions, it's helpful to think of each term as contributing distinct powers or products, and use combinatorics to count unique combinations efficiently.

Learn how to calculate the number of distinct terms in the expansion of (1 + x³ + x⁴)⁸. This solution involves the use of multinomial expansion and combinatorics, specifically the Stars and Bars method, and is suitable for Grades 11-12.

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