Honors Mathematics I

Let A be the set {1, 2, 3, 4, 5}. Are the following true or false:

1. 2 A.
2. {2, 4, 6} A.
3. A {2, 4, 6}.
4. A {1, 2, 3}.
5. {1, 2, 3} A.
6. 7 A.

On a Venn diagram with sets A, B, and C, shade the regions representing:

1. A B C.
2. A^c B C.

Let U be the set of all employees of a hospital. Let:
A = { x U | x is an administrator}
D = { x U | x is a doctor}
F = { x U | x is a female}
M = { x U | x is a male}
N = { x U | x is a nurse}
Describe in words, the following sets:

1. N D.
2. D^c F.
3. A N.

Write the following sets symbolically:

1. The set of all male employees who are administrators.
2. The set of all employees who are administrators or doctors.
3. The set of all employees who are not nurses and are female.

Simplify the set expression A (B C)^c.

Suppose that |A| = 30, |B| = 20, and |A B| = 10. Find |A B|.

To help plan the number of meals to be prepared in a college cafeteria, a survey was conducted and the following data were obtained:

• 130 students ate breakfast in the cafeteria
• 180 students ate lunch in the cafeteria
• 275 students ate dinner in the cafeteria
• 68 students ate breakfast and lunch in the cafeteria
• 112 students ate breakfast and dinner in the cafeteria
• 90 students ate lunch and dinner in the cafeteria
• 58 students ate all three meals in the cafeteria

How many students:

1. ate at least one meal in the cafeteria?
2. ate exactly one meal in the cafeteria?
3. ate only dinner in the cafeteria?
4. ate exactly two meals in the cafeteria?

A traveller, wishing to pack light, decides to take 4 shirts, 3 pants, 2 sports jackets and 5 ties, all with complimentary colours, on an extended trip. How many different ways can these be combined into different outfits (assuming that all 4 types of clothing are worn in an outfit)?

If the traveller decides to not wear a jacket some days, how many different outfits are there now?

Evaluate:

1. 10!
2. P(10,3)
3. C(10,3)
4. P(n, n-3)

How many different ways can the letters of the word "CALYPSO" be arranged? How many different ways can the letters of the word "MURRUMBIDGEE" be arranged?

A bridge hand consists of 13 hands dealt from a deck of 52 cards. How many different bridge hands are there? How many different bridge hands have 5 spades in them? How many bridge hands consist of 5 spades, 4 hearts, 4 diamonds, and 4 clubs?

In men's tennis a match is won by the first player to win 3 sets. How many different ways can the sets of a tennis match be won (the order that the sets are won is important for this problem)? Try to calculate this without listing every possibility.

In poker, a straight flush is a sequence of 5 cards of the same suit in order (eg. 10S, JS, QS, KS, AS), while a straight is a sequence of 5 cards in order, but not all of the same suit (eg. AS, 2S, 3D, 4H, 4S). Aces can be either high or low.

1. How many different hands are straight flushes?
2. How many different hands are flushes (but not straight flushes)?

Three children, Bruce, Sheila and Kevin, earn $10 from a lemonade stand. They decide to split the money amongst themselves in whole-dollar amounts, and each gets at least $1. How many different ways can the money be split amongst the three? (Hint: think of the money laid out in a line, and ask yourself where the divisions between the amounts can be put)

In general, if you have n identical objects and you want to partition them into r distinguished groups, each containing at least one object, how many ways can this be done?