Combinations of Cards
Probability
Level
3
There is a standard deck of 52 cards in front of you, but each card has no suit, and cards of the same number or face are exactly the same. How many combinations of cards are possible when you randomly draw 3 cards from this deck?
The answer is 455.
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We can split this problem into 3 cases:
All 3 cards are matching
2 cards are matching
No cards are matching
If all 3 cards are matching, there are 13 combinations of cards:
A, A, A
2, 2, 2
3, 3, 3
...
K, K, K
If 2 cards are matching, there are 13 choices for the 2 cards that are matching, which leaves 12 choices for the card that isn't matching. We can multiply these amounts together to find how many combinations there are when 2 cards are matching. 13*12= 156 combinations .
If no cards are matching, there are 13 C 3, or 3 ∗ 2 ∗ 1 1 3 ∗ 1 2 ∗ 1 1 combinations, which evaluates to 286 combinations .
We can add the number of combinations in each case together to find the total amount of combinations. 13+156+286= 455 total combinations