A bridge hand consists of 13 cards. How many bridge hands include 5 cards of one suit, 6 cards of a second suit and 2 cards of a third suit?
To find the number of bridge hands that include 5 cards of one suit, 6 cards of a second suit, and 2 cards of a third suit, we can use combinatorics.
First, let's choose the 5 cards for the first suit. Since there are 13 cards in a suit, we need to choose 5 out of the 13 cards. This can be done using the combination formula, denoted as "nCr", where n is the total number of cards and r is the number of cards we want to choose. So, the number of ways to choose 5 cards out of 13 is:
C(13, 5) = 13! / (5!(13 - 5)!) = 1287
Next, we need to choose the 6 cards for the second suit. We have 13 - 5 = 8 remaining cards of this suit. Thus, the number of ways to choose 6 cards out of 8 is:
C(8, 6) = 8! / (6!(8 - 6)!) = 28
Finally, we have only 2 remaining cards for the third suit, so there's only one possible combination.
To find the total number of bridge hands with 5 cards of one suit, 6 cards of a second suit, and 2 cards of a third suit, we multiply the three individual choices together:
Total number of bridge hands = 1287 * 28 * 1 = 36036
Therefore, there are 36,036 bridge hands that include 5 cards of one suit, 6 cards of a second suit, and 2 cards of a third suit.