a bridge hand consists of 13 cards. how many bridge hands include 5 cards of one suit, 6 of a second, and 2 cards of a third?
Very few.
And that had contains a void.
To find the number of bridge hands that include 5 cards of one suit, 6 of a second suit, and 2 cards of a third suit, we can use combinations.
First, let's determine the number of ways to choose 5 cards from one suit. Since there are 13 cards in a suit, we can calculate this using the combination formula, denoted as "nCr":
C(13, 5) = 13! / (5!(13-5)!) = 13! / (5! * 8!) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1) = 1287
Next, we need to count the number of ways to choose 6 cards from the second suit. Similar to above, we can calculate this using combinations:
C(13, 6) = 13! / (6!(13-6)!) = 13! / (6! * 7!) = (13 * 12 * 11 * 10 * 9 * 8) / (6 * 5 * 4 * 3 * 2 * 1) = 1716
Finally, we need to choose 2 cards from the third suit:
C(13, 2) = 13! / (2!(13-2)!) = 13! / (2! * 11!) = (13 * 12) / (2 * 1) = 78
Since each of the above steps is independent of one another, we can multiply them together to get the total number of bridge hands that meet the given criteria:
1287 * 1716 * 78 = 144,187,992
Thus, there are 144,187,992 bridge hands that include 5 cards of one suit, 6 of the second suit, and 2 cards of the third suit.