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how any ways are there to draw a 5 card hand with 3 kings and 2 aces?

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52 card deck

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  • 8 years ago
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    Start with the aces. There are 4 possibilities for the 1st ace. Then there are 3 possibilities for the second ace. So that's a total of 4*3 possibilities. BUT it doesn't make any difference which order you actually draw the aces in. Ace of spades and ace of hearts is the same as ace of hearts and ace of spades. so you have to divide by 2 to get 6 unique pairs of aces.

    Now to the kings.

    4*3*2 = the total number of possibilities. But again it doesn't matter what order we draw them. In this case there are 3 possible orders. Think about it this way. You've got 3 kings and you can put each king in one of three positions. So there are 3*2*1 ways of ordering the kings. So the total number of unique possibilities is:

    (4*3*2)/(3*2*1) = 4.

    So the total number of ways of drawing 3 kings and 2 aces is 6*4 = 24.

    By the way, an alternate way of looking at the kings is this. You have 4 kings in the deck. How many ways are the to exclude 1 king to leave the other 3. Obviously any one of the 4 kings can be excluded which means there are 4 ways of selecting the other three kings.

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