(problem 1) In how many ways can a 5 person committee be formed from a group of
25 employees?
We define and enumerate combinations.
Combinations
Combinations are counted by counting permutations and then dividing appropriately.
example 1 In how many ways can a 6 person committee be formed from a department
of 30 faculty members?
The order in which the members are selected is not important, so the answer is the number of combinations of 30 objects taken 6 at a time, denoted . To count this, we note that each 6 person committee can have its members permuted in ways. Thus . Hence,
The order in which the members are selected is not important, so the answer is the number of combinations of 30 objects taken 6 at a time, denoted . To count this, we note that each 6 person committee can have its members permuted in ways. Thus . Hence,
example 2 How many subsets of size 8 can be formed from a set of size 12?
Since the order of selection of the 8 elements in the subset is not important, the answer is the number of combinations of 12 objects taken 8 at a time, denoted . To count this, we note that each 8 element subset can have its members permuted in ways. Thus .
Since the order of selection of the 8 elements in the subset is not important, the answer is the number of combinations of 12 objects taken 8 at a time, denoted . To count this, we note that each 8 element subset can have its members permuted in ways. Thus .
- Proof
- Each combination of the objects selected from a set of distinct objects corresponds to permutations of those objects. This is because there are ways to permute the selected objects. Thus the number of permutations, is times greater than the number of combinations, . Since we know the number of permutations is we have Dividing both sides by gives the desired result, namely
example 4 An NBA basketball team consists of 15 players. In how many
ways can a coach choose the starting 5 (without regard to position, ability,
etc)?
The order in which the 5 players are selected is not important, so the answer is .
The order in which the 5 players are selected is not important, so the answer is .
(problem 4) An NHL hockey team consists of 23 players. A line-up consists of
6 players. How many line-ups are possible (without regard to position)?
(problem 5) How many subsets of size can be made from a set containing elements
()?
How does this answer differ from the answer to the example above?
How does this answer differ from the answer to the example above?
example 6 A box of crayons contains 64 different colors. In how many ways can Jack
select 6 different colors to make his drawing?
Since the order of selection of the 6 colors is not important, there are ways to select the 6 colors from a box containing 64 different colors.
Since the order of selection of the 6 colors is not important, there are ways to select the 6 colors from a box containing 64 different colors.
(problem 6) A box of crayons contains 64 different colors. In how many ways can Jill
select 10 different colors to make her drawing?
example 7 An ordinary deck of playing cards contains 52 cards broken into two colors,
red and black (26 of each). Each color is further subdivided into two suits (of 13
cards each)- red are evenly split between hearts and diamonds, black between
spades and clubs. Finally, each suit is decomposed into 13 types labeled 2
through 10, then Jack, Queen, King and Ace. In the popular card game
poker, each player is dealt 5 cards. How many different 5 card hands are
possible?
The answer is simply since the order in which the cards are received is irrelevant (and quickly becomes unknown when the holder organizes the cards in their hand).
The answer is simply since the order in which the cards are received is irrelevant (and quickly becomes unknown when the holder organizes the cards in their hand).
(problem 7) There are variants of poker in which a player is dealt 7 cards. How many
-card poker hands are possible?
example 8 In poker, one type of hand is called “three of a kind”. This occurs when 3
of your 5 cards are of the same type and the remaining two cards are of two
other types. For example, a hand consisting of 3 Jacks, an Eight and a Four
(regardless of suit) is a three of a kind (3 Jacks). How many three of a kinds are
possible?
We can break this down into a two step process. First we choose the 3 cards to make the “three of a kind”, then choose the remaining two cards. To choose the 3 matching cards, there are 13 choices for the type and then there are ways to choose the suits. To choose the remaining two cards, we first choose two different types in ways (one of the 13 types is accounted for, so there are only 12 left), and then choose the suit of each card in ways. Putting all of this together, the number of ”three of a kind’s” is
We can break this down into a two step process. First we choose the 3 cards to make the “three of a kind”, then choose the remaining two cards. To choose the 3 matching cards, there are 13 choices for the type and then there are ways to choose the suits. To choose the remaining two cards, we first choose two different types in ways (one of the 13 types is accounted for, so there are only 12 left), and then choose the suit of each card in ways. Putting all of this together, the number of ”three of a kind’s” is