From a standard deck of 52 playing cards, 3 cards are drawn together at random.\nWhat is the probability that all the three cards drawn are kings?

Introduction / Context:
This question checks understanding of basic probability with combinations. We are drawing several cards simultaneously from a standard deck and must find the probability that a specific rare pattern occurs, namely that every card drawn is a king. This uses the idea of favourable combinations divided by total combinations.

Given Data / Assumptions:

• Standard deck size = 52 cards.
• Number of kings in the deck = 4.
• Number of cards drawn at random = 3, drawn together without replacement.
• All 3 cards must be kings.
• All 3 card subsets of the deck are equally likely outcomes.

Concept / Approach:
When each outcome is a set of cards with equal probability, the probability is computed as:\nProbability = (number of favourable combinations) / (total number of possible combinations).\nHere, combinations C(n,r) are used because the order of cards in a hand does not matter. We count how many ways to choose 3 kings from the 4 available, and how many ways to choose any 3 cards from the 52 cards in the deck.

Step-by-Step Solution:
Total ways to choose 3 cards from 52 = C(52,3). Compute C(52,3) = 52 * 51 * 50 / (3 * 2 * 1) = 22100. Favourable ways: choose 3 kings from the 4 kings in the deck. Number of favourable ways = C(4,3) = 4. Probability that all three cards are kings = C(4,3) / C(52,3) = 4 / 22100. Simplify 4 / 22100 by dividing numerator and denominator by 4 to get 1 / 5525.

Verification / Alternative check:
We can also compute the same probability stepwise. The probability that the first card drawn is a king is 4 / 52. Given that, the probability that the second is also a king is 3 / 51, and the probability that the third is also a king is 2 / 50. Multiply these: (4 / 52) * (3 / 51) * (2 / 50). Simplify the numerators and denominators step by step, and you will again arrive at 1 / 5525. This confirms that the combination method is consistent with the sequential probability approach.

Why Other Options Are Wrong:

• 1/5225: Very close but incorrect; it shows a small arithmetic slip in computing combinations.
• 5525: This is not even a probability value and is clearly invalid, since probabilities must be between 0 and 1.
• 1/525: Too large; this would suggest the event is much more likely than it really is and does not match correct combinatorial counts.

Common Pitfalls:
Many students inadvertently treat the draws as ordered and then forget to adjust, or they mix up combinations and permutations. Another frequent mistake is failing to reduce the fraction correctly, leading to values like 4 / 22100 instead of the simplified 1 / 5525. Some also misread the question and compute the probability of "at least one king" rather than "all three kings". Careful reading and a clear distinction between ordered and unordered counting are the keys here.

Final Answer:
The probability that all three cards drawn are kings is 1/5525.