From a deck of 52 cards, 3 cards are drawn randomly. What is the probability…

Concept: When cards are drawn together (an unordered selection), probability is found using combinations — Probability = (favourable outcomes) / (total outcomes), with each count worked out using nCr. When the favourable event needs exactly one card from each of several independent categories, the favourable count is the product of choosing one item from each category using nC1 for that category's size.

1. Total ways to draw any 3 cards from the 52-card deck: 52C3.

2. Compute 52C3 = (52×51×50)/(3×2×1) = 22100.

3. Ways to pick the required spade card from the 13 spades: 13C1 = 13.

4. Ways to pick the required red queen from the 2 red queens (hearts and diamonds): 2C1 = 2.

5. Ways to pick the required black king from the 2 black kings (spades and clubs): 2C1 = 2.

6. Multiply the three category counts to get the favourable outcomes: 13 × 2 × 2 = 52.

7. Probability = favourable / total = 52 / 22100.

Cross-check: dividing both terms of 52/22100 by 52 gives 22100 ÷ 52 = 425 exactly, so the fraction reduces cleanly to 1/425 with no remainder, confirming the simplification.

So the required probability is 1/425.