The mythical Round Table of King Arthur. Note that the table indicates where King Arthur sat as well as where 24 knights sat. In how many ways can n people be seated around a similar table if:

a. a head of the table is designated?

b. no head of the table is designated?

The tree diagram shows the four possible outcomes when a coin is tossed twice. If H and T represent "heads" and "tails," respectively, then the four outcomes are HH, HT, TH, and TT.

a. Make a tree diagram showing the outcomes if a coin is tossed 3 times.

b. The solid lines of the tree diagram are called branches and the elements of the bottom row (H, T, H, and T) are called leaves. How many branches and leaves are in your diagram for part (a)?

c. How many branches and leaves would there be in a tree diagram showing the toss of a coin 10 times?

a. A hiker would like to invite 7 friends to go on a trip but has room for only 4 of them. In how many ways can they be chosen?

b. If there were room for only 3 friends, in how many ways could they be chosen? How is your answer related to the answer for part (a)? Why?

Give the expansion of the binomial. Simplify your answer.

$$\left(x+\frac{1}{x}\right)^6$$

In how many ways can 4 different prizes be given to any 4 of 10 people if no person receives more than 1 prize?

Let U = the universal set of all teachers in your school. Let the subsets of mathematics teachers, biology teachers, physics teachers, and chemistry teachers be represented by M, B, P, and C, respectively. Describe in words the following set, and name a teacher belonging to each set if such a teacher exists in your school.

a. $\bar{M} \cap \bar{B} \cap \bar{P} \cap \bar{C}$

b. $\overline{M \cup B \cup P \cup C}$

If A is any subset of a universal set U, complete the following.

a. $A \cup \phi=$ ?

b. $A \cap \phi=$ ?

c. $A \cup \bar{A}=$ ?

d. $A \cap \bar{A}=$ ?

e. $A \cup U=$ ?

f. $A \cap U=$ ?

Give the expansion of the binomial. Simplify your answer.

$$\left(1+\frac{x}{2}\right)^3$$

A person bicycles along the city streets by always traveling south and east.

To the nearest tenth of a percent, what percent of the routes from X to Y pass through P?

Of the 12 players on a school's basketball team, the coach must choose 5 players to be in the starting lineup. In how many ways can this be done if the playing positions are:

(a) considered?

(b) not considered?

A radio show plays the top 10 musical hits of the previous week. During the first full week of last January, these 10 hits were chosen from 70 possibilities, and they were played in order of increasing popularity. How many possible orders were there? Note: Since calculators cannot evaluate large factorials, use the approximation given by the formula $n ! \approx \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$.

a. In how many ways can a club with 20 members choose a president and a vice president?

b. In how many ways can the club choose a 2-person governing council?

Study Pascal's triangle. Describe the symmetry that the numbers in the triangle have.

Give the expansion of the binomial. Simplify your answer.

a. $(a+b)^3$

b. $(20+1)^3$

c. $(20-1)^3$