To find the number of terms in the expansion of the expression \((x+a)^{20} + (x-a)^{20} + (x+ai)^{20} + (x-ai)^{20}\), we can break it down step by step.
### Step 1: Analyze the first two expansions
Consider the first two terms: \((x+a)^{20}\) and \((x-a)^{20}\).
Using the Binomial Theorem, the general term in the expansion of \((x + a)^{20}\) is given by:
\[
T_k = \binom{20}{k} x^{20-k} a^k
\]
Similarly, for \((x - a)^{20}\), the general term is:
\[
T_k' = \binom{20}{k} x^{20-k} (-a)^k = \binom{20}{k} x^{20-k} (-1)^k a^k
\]
When we add these two expansions:
\[
(x+a)^{20} + (x-a)^{20}
\]
the terms where \(k\) is odd will cancel out, while the terms where \(k\) is even will remain. Therefore, we only consider even \(k\).
### Step 2: Identify the even terms
The even values of \(k\) from \(0\) to \(20\) are \(0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20\). This gives us a total of:
\[
\frac{20}{2} + 1 = 11 \text{ terms}
\]
### Step 3: Analyze the second two expansions
Now consider the next two terms: \((x+ai)^{20}\) and \((x-ai)^{20}\).
Using the same reasoning, the general term in the expansion of \((x + ai)^{20}\) is:
\[
T_k'' = \binom{20}{k} x^{20-k} (ai)^k
\]
And for \((x - ai)^{20}\):
\[
T_k''' = \binom{20}{k} x^{20-k} (-ai)^k = \binom{20}{k} x^{20-k} (-1)^k (ai)^k
\]
When we add these two expansions:
\[
(x+ai)^{20} + (x-ai)^{20}
\]
the terms where \(k\) is odd will again cancel out, while the terms where \(k\) is even will remain.
### Step 4: Identify the even terms for \(ai\)
Similar to the previous case, the even values of \(k\) from \(0\) to \(20\) are \(0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20\). This also gives us:
\[
\frac{20}{2} + 1 = 11 \text{ terms}
\]
### Step 5: Combine the results
Now we combine the results from both parts:
- From \((x+a)^{20} + (x-a)^{20}\), we have 11 terms.
- From \((x+ai)^{20} + (x-ai)^{20}\), we also have 11 terms.
However, we need to check for any overlapping terms. The terms from both expansions are of the form \(x^{20-2k}a^{2k}\) and \(x^{20-2k}(ai)^{2k}\). Since \(a^{2k}\) and \((ai)^{2k}\) are distinct (as \(i\) introduces a different factor), there are no overlapping terms.
### Conclusion
Thus, the total number of distinct terms in the entire expression is:
\[
11 + 11 = 22
\]
### Final Answer
The total number of terms in the expansion of \((x+a)^{20} + (x-a)^{20} + (x+ai)^{20} + (x-ai)^{20}\) is **22**.
