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 determine the number of bridge hands that contain each of the given patterns, we need to use the principles of combinations.
a.) Eight Spades and five Diamonds:
In a standard deck of 52 cards, there are 13 Spades and 13 Diamonds. We need to choose 8 Spades from the 13 available and 5 Diamonds from the 13 available. The number of ways to choose these cards can be calculated using the combination formula.
The number of ways to choose 8 Spades from 13 is given by C(13, 8), and the number of ways to choose 5 Diamonds from 13 is given by C(13, 5). To find the number of bridge hands containing this pattern, we multiply these two combinations together.
The combination formula is given by C(n, r) = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items chosen at a time.
So, the number of bridge hands containing eight Spades and five Diamonds is:
C(13, 8) * C(13, 5) = (13! / (8! * (13-8)!)) * (13! / (5! * (13-5)!))
b.) An 8-card suit and a 5-card suit:
In this case, we need to choose any 8 cards from a suit and any 5 cards from another suit. There are 4 suits in a deck (Spades, Hearts, Diamonds, and Clubs).
The number of ways to choose 8 cards from one suit is given by C(13, 8), and the number of ways to choose 5 cards from another suit is given by C(13, 5). However, we also have to consider which suits we choose for each pattern. We can choose 2 suits for these patterns in C(4, 2) ways.
So, the number of bridge hands containing an 8-card suit and a 5-card suit is:
C(4, 2) * (C(13, 8) * C(13, 5))
To get the final answers, simply calculate these combinations using their respective formulas.