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Question:

You ask for the type of item in box 45. Instead of being given a direct answer, you are told that there are 31 items of the same type as box 45 in boxes 1 to 44 and 43 items of the same type as box 45 in boxes 46 to 100.

What is the maximum possible number of different types of items?

Explanation:

Considering the given options, the maximum number of different types can be 6.

Assume that there are 6 items.

Now number of items of same type as the one in box 45 = 1 + 31 + 43 = 75

So number of remaining items = 25

1 + 2 + 4 + 8 + 16 = 31. If there are 5 types of items, the minimum number of items of 5 types = 31.

31 + 75 >100

So, there cannot be 6 types of items.

Now consider that there are 5 types of items.

Now number of items of same type as the one in box 45 = 1 + 31 + 43 = 75

So number of remaining items = 25

Now, 25 = (1 + 2 + 4 + 17) or (1 + 3 + 6 + 16) ,… etc

So there can be 5 types of different items.

Hence, option (a).