To determine the number of distinct terms in the expansion of $(x + y)^{100}$ after collecting like terms, we use the concept of binomial expansion.

The general expansion of $(x + y)^n$ is given by: $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$

Each term in the expansion is of the form $x^{n-k} y^k$, where $k$ ranges from 0 to $n$. For each unique pair of powers $(n - k, k)$, we get a distinct term. Here, since $n = 100$, $k$ can range from 0 to 100, providing 101 possible terms (one for each integer value of $k$).

Thus, there are 101 distinct terms in the expansion of $(x + y)^{100}$ after like terms are collected.

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Here are some related questions:

1. How can we calculate specific coefficients in the binomial expansion of $(x + y)^{100}$?
2. What is the general form of a term in a binomial expansion?
3. How does the binomial theorem apply to cases where $x$ or $y$ has a coefficient other than 1?
4. Can we determine the middle term in the expansion of $(x + y)^{100}$?
5. How does the binomial expansion apply to negative or fractional exponents?

Tip: In a binomial expansion, the number of distinct terms is always $n + 1$ for $(x + y)^n$, because each term corresponds to a unique power combination of $x$ and $y$.