To solve the problem of determining how many hands of 6 cards in cribbage contain at least two queens, we can break it down as follows:

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Step 1: Total number of hands

The total number of ways to deal a hand of 6 cards from a 52-card deck is given by: $\binom{52}{6} = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 20,358,520$

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Step 2: Find the complement ("no queens" and "exactly one queen")

Let’s use the complement rule:

• "At least two queens" is the same as not having 0 queens or 1 queen.

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Step 3: Calculate hands with 0 queens

• There are 4 queens in a deck of 52 cards, so the number of cards excluding queens is 48.
• The number of hands with 0 queens is given by selecting all 6 cards from the 48 non-queen cards: $\binom{48}{6} = \frac{48 \times 47 \times 46 \times 45 \times 44 \times 43}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 12,271,512$

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Step 4: Calculate hands with exactly 1 queen

• We must select 1 queen out of the 4 available:
  $\binom{4}{1} = 4$

• Then we must select the remaining 5 cards from the 48 non-queen cards: $\binom{48}{5} = \frac{48 \times 47 \times 46 \times 45 \times 44}{5 \times 4 \times 3 \times 2 \times 1} = 1,712,304$

• Therefore, the number of hands with exactly 1 queen is: $\binom{4}{1} \times \binom{48}{5} = 4 \times 1,712,304 = 6,849,216$

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Step 5: Apply the complement rule

The number of hands with at least two queens is given by: $\text{Total hands} - (\text{Hands with 0 queens} + \text{Hands with exactly 1 queen})$ $= 20,358,520 - (12,271,512 + 6,849,216) = 20,358,520 - 19,120,728 = 1,237,792$

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Conclusion:

The number of hands containing at least two queens is: $\boxed{1,237,792}$

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Would you like further details on this or another explanation?

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Here are some related questions:

1. What is the probability of getting at least two queens in a cribbage hand?
2. How many hands contain exactly three queens?
3. How many hands contain all four queens?
4. What is the probability of having no queens in a 6-card hand?
5. How does this change if we deal a 5-card hand instead?

Tip: The complement rule is a powerful technique for calculating probabilities or counting cases without directly enumerating complex conditions.