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?

(a) 27   

(b) 36

(c) 44 

(d) 45

Raj has 10 pairs of red shoes, 9 pairs of white shoes, and 8 pairs of black shoes. In total, he has:

- 20 red shoes

- 18 white shoes

- 16 black shoes

In the worst-case scenario, Raj might pick all the shoes that do not form a red pair before he finally picks a pair of red shoes.

Detailed Analysis:

1. Picking all non-red shoes first: 

   - There are 34 non-red shoes (18 white and 16 black).

2. Picking individual red shoes without forming a pair:

   - Out of the 20 red shoes, if Raj picks 1 shoe from each of the 10 pairs first, he will have picked 10 red shoes without forming a pair.

In this scenario, he could have picked:

- All 34 non-red shoes.

- 10 red shoes (one from each pair without completing a pair).

After picking these 44 shoes (34 non-red + 10 single red shoes), the next shoe he picks must form a pair with one of the previously picked 10 red shoes.

Therefore, the maximum number of attempts Raj needs to make to ensure he gets a pair of red shoes in the worst-case scenario is:

34 (non-red shoes) + 10 (single red shoes) + 1 (to complete the red pair) = 45 attempts

Thus, the maximum number of attempts Raj may need to make to get a pair of red shoes is 45.

Correct Answer: (d) 45

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