From a standard deck of 52 playing cards, a 5 card hand is dealt.\nIn how many distinct ways can the hand contain at least 2 queens?

Introduction / Context:
This question asks for the number of 5 card hands that contain at least two queens. It is a combinatorics problem involving casework on how many queens appear. We need to carefully consider separate situations for exactly 2 queens, exactly 3 queens and all 4 queens, and then add the counts.

Given Data / Assumptions:

• Standard deck: 52 playing cards.
• Total queens in the deck = 4.
• Hand size = 5 cards.
• We need hands with at least 2 queens, meaning 2, 3 or 4 queens are allowed.
• Order of cards in the hand does not matter.

Concept / Approach:
We use combinations for hands. For each allowed number of queens k (where k = 2, 3 or 4), we choose which k queens appear using C(4,k), and then fill the remaining 5 - k places with non queen cards chosen from the 48 remaining cards. The total count is the sum over these three cases. This approach avoids double counting, because each hand has a unique number of queens.

Step-by-Step Solution:
Case 1: Exactly 2 queens in the hand. Choose 2 queens from 4: C(4,2) = 6. Remaining cards needed = 5 - 2 = 3 non queens from 48 others: C(48,3). Case 1 count = C(4,2) * C(48,3). Case 2: Exactly 3 queens in the hand. Choose 3 queens from 4: C(4,3) = 4. Remaining cards needed = 5 - 3 = 2 non queens from 48 others: C(48,2). Case 2 count = C(4,3) * C(48,2). Case 3: All 4 queens in the hand. Choose all 4 queens: C(4,4) = 1. Remaining one card from 48 non queens: C(48,1) = 48. Case 3 count = C(4,4) * C(48,1) = 48. Now compute the values: C(48,3) = 48 * 47 * 46 / (3 * 2 * 1) = 17296. So Case 1 count = 6 * 17296 = 103776. C(48,2) = 48 * 47 / 2 = 1128. So Case 2 count = 4 * 1128 = 4512. Case 3 count = 48. Total hands with at least 2 queens = 103776 + 4512 + 48 = 108336.

Verification / Alternative check:
We can compare with the total number of 5 card hands, which is C(52,5) = 2598960. The required hands (108336) form a small but plausible fraction of all possible hands. Also, the dominant contribution is from the case of exactly 2 queens, which intuitively should be the most common among hands with at least two queens. The arithmetic for each combination has been broken into small steps to minimise mistakes and the final sum has been verified step by step.

Why Other Options Are Wrong:

• 103336: Slightly smaller than the correct value and likely comes from counting only the 2 queen case and then adding an incorrect figure for the higher cases.
• 120000: Too large for this specific requirement and not equal to the precise combination sum.
• 108333: Very close but not exact; it indicates a small addition or subtraction error in combining the cases.

Common Pitfalls:
Students sometimes misread "at least 2 queens" as "exactly 2 queens" and ignore the 3 and 4 queen hands. Others double count by not splitting the problem into disjoint cases clearly. It is also easy to make arithmetic mistakes when computing large combinations, so writing numerators and denominators explicitly and simplifying carefully is important. Always ensure that your final answer is the sum of all valid, non overlapping cases.

Final Answer:
The number of 5 card hands that contain at least two queens is 108336.