Answer: Approximately 0.00004597583714
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Step-by-step explanation:
Four-of-a-kind is when you get four cards of the same value. One example is if we got four aces. Any four-of-a-kind already has two pairs built into it. I'm assuming your teacher wants four-of-a-kind and another different pair (we'll have 6 pairs all together).
In any suit there are 13 unique cards. So there are 13 choices to fill the first four slots to set up the four-of-a-kind.
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After the four-of-a-kind is formed, we have 13-1 = 12 unique cards left in any given suit. Once that fifth card is chosen, we then have 4 C 2 = 6 ways to select that final card such that we get another pair. I'm using the nCr combination formula since order doesn't matter. The steps to calculating this value (and the other nCr value mentioned later) is shown in the attached image below.
Multiplying those values gets us 13*12*6 = 936 different six card hands such that we get four-of-a-kind and a different pair.
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This is out of 52 C 6 = 20,358,520 different six card hands.
Divide the two values found
936/(20,358,520) = 0.00004597583714