Raj has ten pairs of red, nine pairs of white and eight pairs of black shoes in a box. If he randomly picks shoes one by one (without replacement) from the box to get a red pair of shoes to wear, what is the maximum number of attempts he has to make?

To find the maximum number of attempts, we consider the worst-case scenario. Raj wants a red pair of shoes (one left red, one right red).

1. First, Raj picks all the shoes that are not red.

   • White shoes: 9 pairs = 18 individual shoes
   • Black shoes: 8 pairs = 16 individual shoes
   • Total non-red shoes picked = 18 + 16 = 34 shoes. At this point, he has no red pair, and only red shoes are left in the box. There are 10 pairs of red shoes, meaning 20 individual red shoes (10 left, 10 right).

2. Next, he starts picking red shoes. To avoid getting a pair for as long as possible, he picks all the red shoes of one type (e.g., all left red shoes).

   • He picks 10 red shoes (all lefts, or all rights).
   • Total shoes picked so far = 34 (non-red) + 10 (red of one type) = 44 shoes. At this point, he has 10 left red shoes in his hand, but no pair, as he hasn't picked any right red shoes yet (or vice-versa). The box now contains only the remaining 10 right red shoes.

3. The very next shoe he picks must be a right red shoe. Since he already has all 10 left red shoes, any right red shoe he picks will immediately form a pair with one of the left red shoes he already has.

   • He picks 1 more shoe.
   • Total attempts = 44 + 1 = 45.

Therefore, the maximum number of attempts he has to make to get a red pair of shoes is 45.

The final answer is D