Binomial Theorem

The given expression is . To find the total number of terms in the expansion of this expression, we will first analyze the individual expansions of and , and then combine them.

Step 1: Expansion of

The binomial expansion of is given by:

This expansion has terms because the index ranges from 0 to 100, inclusive. So, the expansion of contains 101 terms.

Step 2: Expansion of

The binomial expansion of is given by:

Like the first expansion, this also has terms. However, the signs of the terms alternate depending on whether is even or odd due to the factor . So, the expansion of also contains 101 terms.

Step 3: Combining the Expansions

Now, consider the sum of the two expansions:

When we add these two expansions, terms where the powers of are odd will cancel each other out, because will have the opposite sign of when is odd. On the other hand, terms where is even will add up, because both expansions will have the same sign for those terms.

Thus, only the terms where is even will remain after addition. The even values of range from 0 to 100, inclusive. These values are , which gives a total of 51 even values.

Final Answer:

The total number of terms in the expansion of is 51, because only the terms with even values of survive after adding the two expansions