Binomial Expansion

To determine the total number of unique terms in the expansion of , we can approach the problem as follows:

The binomial expansion of and are:

• :
• : , which is

The terms in these expansions depend on the parity of (even or odd):

• For even: Both and contribute the terms .
• For odd: The terms cancel each other out because of the opposite signs from and .

This results in non-zero terms only when is even. Since can go from 0 to 60, the sequence of even numbers between 0 and 60 is .

The number of even numbers in this sequence can be found by computing the number of terms in an arithmetic sequence:

The sequence has a common difference of 2. The formula for the number of terms in an arithmetic sequence is calculated as:

Substituting the relevant values, we find:

Thus, the total number of terms in the expansion of is 31.