Which of the following is not possible?

The correct option is 2.

Let the number of items of types a, b, c, d, e, etc., be denoted by N(a), N(b), N(c), N(d), N(e), etc.
We are given that there are 100 boxes in total, and the number of items of each type must at least double as you move to the next type, with N(a) = 1.
Therefore:
N(b) ≥ 2 × N(a) = 2
N(c) ≥ 2 × N(b) ≥ 4
N(d) ≥ 2 × N(c) ≥ 8
N(e) ≥ 2 × N(d) ≥ 16

Let us test the option where there are exactly 45 items of type c (N(c) = 45):
1. If N(c) = 45, then the next type (type d) must have at least twice as many items as type c:
N(d) ≥ 2 × 45 = 90.
2. The sum of the counts for types c and d alone would then be:
N(c) + N(d) ≥ 45 + 90 = 135.
3. This exceeds the total limit of 100 boxes.

Even if type c were the last type of item (meaning no type d exists), the maximum number of items for types a, b, and c would be:
N(a) + N(b) + N(c) ≤ 1 + 22 + 45 = 68.
This is strictly less than 100, meaning we would have leftover boxes that cannot be filled without introducing another type or violating the doubling rule.

Thus, it is mathematically impossible to have exactly 45 items of type c.