(problem 1a) Two 6-sided dice are thrown, one of which is red and the other green. In
how many ways can their sum be at most 5?
The number of outcomes whose sum is at most 5 is .
We use partitions to enumerate sets.
1 Partitions
When counting the number of elements in a set, it is sometimes helpful to break the set into distinct pieces and count the number of elements in each piece.
Partition A partition of a set is a collection of non-empty sets satisfying the
following two properties: \begin{align*} 1) & S_i \cap S_j = \emptyset \text{ for } i \neq j \text{ with } 1 \leq i,j \leq n\\ 2) & S_1 \cup S_2 \cup \cdots \cup S_n = S \end{align*}
We will denote the number of elements in a set by . The main result is that the number of elements in a set is the sum of the elements in each subset of the partition: where is a partition of .
example 1 Two 6-sided dice are thrown, one of which is red and the other green. In
how many ways can their sum be at least 10?
We have seen that the total number of outcomes of the two dice when recorded as (red, green) is 36. How many of these 36 possible outcomes produce a sum of 10 or higher (10, 11 or 12)? If we let be the set of all outcomes whose sum is greater than or equal to 10, then we can partition into three subsets: and where the subscript identifies the sum. The set contains the outcomes and , so . Verify that and .
Finally, since is a partition of , we have Thus there are 6 ways to get a sum of at least 10 when rolling a pair of 6-sided dice.
We have seen that the total number of outcomes of the two dice when recorded as (red, green) is 36. How many of these 36 possible outcomes produce a sum of 10 or higher (10, 11 or 12)? If we let be the set of all outcomes whose sum is greater than or equal to 10, then we can partition into three subsets: and where the subscript identifies the sum. The set contains the outcomes and , so . Verify that and .
Finally, since is a partition of , we have Thus there are 6 ways to get a sum of at least 10 when rolling a pair of 6-sided dice.
(problem 1b) Two 6-sided dice are thrown, one of which is red and the other green. In
how many ways can their sum be a prime number?
The number of outcomes whose sum is prime is .
The number of outcomes whose sum is prime is .
(problem 1c) Two 6-sided dice are thrown, one of which is red and the other green. In
how many ways can their sum be an odd number?
The number of outcomes whose sum is odd is .
The number of outcomes whose sum is odd is .
example 2 Steph has three different shirts, two different pairs of pants and 14
different hats. How many ways can Steph dress if the hat is optional?
If we denote by the set of all possible outfits Steph can wear, then we can partition into those outfits with a hat, , and those without a hat, . The number of outfits with a hat is and the number of outfits with no hat is . Thus, the total number of ways for Steph to dress is Note that we could have counted this slightly differently, by including “no hat” as a option for the hat: .
If we denote by the set of all possible outfits Steph can wear, then we can partition into those outfits with a hat, , and those without a hat, . The number of outfits with a hat is and the number of outfits with no hat is . Thus, the total number of ways for Steph to dress is Note that we could have counted this slightly differently, by including “no hat” as a option for the hat: .
(problem 2) Jake is going jogging. Jake has 4 t-shirts, 3 pairs of shorts and 2 pairs of
sneakers. How many jogging outfits can Jake make consisting of a pair of shorts, a
pair of sneakers and an optional t-shirt?
example 3 Sam wants to buy a slice of pizza. Sam enjoys three styles of pizza:
Neapolitan, Chicago and Sicilian, and either eats the pizza plain or with pepperoni.
How many different slices of pizza can Sam order?
If we denote by the set of all possible acceptable slices of pizza, then we can partition into the slices that have pepperoni, and those that do not, . The number of slices of each is 3 and hence, the total number of possible slices Sam can order is
If we denote by the set of all possible acceptable slices of pizza, then we can partition into the slices that have pepperoni, and those that do not, . The number of slices of each is 3 and hence, the total number of possible slices Sam can order is
(problem 3) A menu consists of 6 appetizers, 10 entrees and 4 desserts. If a meal
consists of an appetizer and an entree with dessert being optional, how many meals
are possible?
example 4 Alex and Charlie want to have 2 or fewer children. How many
combinations of boys and girls are possible assuming that birth order of each gender
is irrelevant?
If is the set of all possible genders of their children, then we can partition according to the number of children. Let be the number of ways for them to have 0 children. Then because the only way to have 0 children is to have 0 boys and 0 girls. Let be the number of ways for them to have 1 child. Then because they can have 1 boy and 0 girls or 1 girl and 0 boys. Let be the number of ways for them to have 2 children. Then because they can have 2 boys and 0 girls, 2 girls and 0 boys or 1 of each. Thus, the total number of combinations of boys and girls that they can have is
(problem 4) Fluffy is about to give birth to either 5 or 6 kittens. How many
combinations of males and females are possible assuming that birth order of each
gender is irrelevant?
The total number of combinations of males and females that Fluffy can have is
The total number of combinations of males and females that Fluffy can have is
example 5 A 20-sided die is rolled, and a card is drawn. If the die roll is a prime
number, then the suit of the card is noted. If the die roll is not prime, then the
type of the card is noted instead. Thus, the outcomes are recorded as either,
(prime number, suit) or (not prime number, type). How many outcomes are
possible?
There are 8 primes on a 20-sided die: 2, 3, 5, 7, 11, 13, 17 and 19. Playing cards come in 4 suits with 13 types in each suit. Let is the set of all possible outcomes and partition according to whether the die roll is prime or not. Let be the outcomes corresponding to a prime die roll and let correspond to the outcomes for which the die roll is not prime. Then by the FPC, and . By the partitioning principle, the total number of outcomes from the die roll and the card draw is
There are 8 primes on a 20-sided die: 2, 3, 5, 7, 11, 13, 17 and 19. Playing cards come in 4 suits with 13 types in each suit. Let is the set of all possible outcomes and partition according to whether the die roll is prime or not. Let be the outcomes corresponding to a prime die roll and let correspond to the outcomes for which the die roll is not prime. Then by the FPC, and . By the partitioning principle, the total number of outcomes from the die roll and the card draw is
(problem 5) A 12-sided die is rolled, and a card is drawn. If the die roll is a prime
number, then the color of the card (red or black) is noted. If the die roll is not prime,
then the suit of the card is noted instead. Thus, the outcomes are recorded as either:
(prime number, color) or (not prime number, suit). How many such outcomes are
possible?
The total number of outcomes is
2025-02-16 03:52:46 The total number of outcomes is