1.) How many bridge hands contain each of the following patterns:
a.) Eight Spades and five Diamonds?
b.) An 8-card suit and a 5-card suit?
(this problem has to do with combination)
To find the number of bridge hands that contain specific patterns, we can use the concept of combinations.
a.) To find the number of bridge hands with eight Spades and five Diamonds, we need to choose 8 cards from the 13 Spades and 5 cards from the 13 Diamonds.
The number of ways to choose 8 cards from 13 Spades is denoted as C(13, 8), and the number of ways to choose 5 cards from 13 Diamonds is denoted as C(13, 5).
Using the combination formula C(n, r) = n! / (r! * (n-r)!), we can calculate:
C(13, 8) = 13! / (8! * (13-8)!) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1) = 1,716
C(13, 5) = 13! / (5! * (13-5)!) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1) = 1,287
Therefore, the number of bridge hands with eight Spades and five Diamonds is the product of these two combinations: 1,716 * 1,287 = 2,208,612.
b.) To find the number of bridge hands with an 8-card suit and a 5-card suit, we need to choose any 8 cards from 13 cards in one suit and any 5 cards from another suit (it can be any suit except the one with 8 cards).
The number of ways to choose 8 cards from 13 is denoted as C(13, 8), and the number of ways to choose 5 cards from 13 is denoted as C(13, 5).
Using the combination formula, we can calculate these combinations:
C(13, 8) = 13! / (8! * (13-8)!) = 1,716
C(13, 5) = 13! / (5! * (13-5)!) = 1,287
Since there are four suits in a deck of cards and we can choose any suit for the 8-card and 5-card suits, we multiply the combinations by 4 (the number of suits) to consider all the possibilities.
Therefore, the total number of bridge hands with an 8-card suit and a 5-card suit is: (1,716 * 1,287) * 4 = 8,832.