Final answer:
The binomial expansion of ((x + y)^{115}) has 116 terms after collecting like terms, based on the binomial theorem which states that a binomial raised to the power n will result in (n + 1) terms.
Step-by-step explanation:
The number of terms in the expansion of a binomial expression (x + y)^n is given by (n + 1), where n is the exponent. After like terms are collected in the binomial expansion of ((x + y)^{115}), there will be 116 terms. This is due to the binomial theorem, which states that when expanding (a + b)^n, the number of terms produced will be one more than the power n, because you have terms from a^n (where b is raised to 0) to b^n (where a is raised to 0), inclusive. Therefore, in a complete expansion, each power of a from n to 0 appears exactly once, as does each corresponding power of b from 0 to n.