SOLUTION: How many different five card hands can be dealt with at least two face cards dealt from a standard deck of playing cards

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Question 1157297: How many different five card hands can be dealt with at least two face cards dealt from a standard deck of playing cards
Answer by KMST(5434) About Me  (Show Source):
You can put this solution on YOUR website!
A standard Anglo-American deck of cards consists of 52 cards divided into 4 suits.
We are not going to consider the jokers, or anything else that many sets of cards may include.
Each suit (hearts, diamonds, spades, and clubs ♥ ♦ ♠ ♣) has 3 face cards (King, Queen, and Jack) and 10 pip cards with between 1 and 10 pips representing the suit.
That makes a total of 52 cards: 12 face cards and 40 pip cards.

From that deck we can make many hands (combinations) of 5 cards.
The number of such combinations is
C%2852%2C5%29=52%2A51%2A50%2A49%2A48%2F%285%2A4%2A3%2A2%29=2598960
We are only interested in the ones with at least 2 pip cards
Some of the 2598960 possible 5-card hands will consist of all face cards.
The number of those would be
C%2840%2C5%29=40%2A39%2A38%2A37%2A36%2F%285%2A4%2A3%2A2%29=658008 .
Some others will have just one of the 12 face cards and one of the combinations of 4 of the 40 pip cards.
There are C%2840%2C4%29=%2840%2A39%2A38%2A37%29%2F%284%2A3%2A2%29=91390 of those 4-pip card sets, so the total number of 5 card hands with exactly one pip card is 12%2A91390=1096680

That leaves us with 2598960-%28658008%2B1096680%29=2598960-1754688=highlight%28844272%29 5-card hands with at least 2 pip cards.

We could also calculate the numbers of 2-pip, 3-pip, 4-pip and 5-pip 5-card hands and add them up to get the same result.