How many circular permutations are possible when seating 5 people around a circular table?

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?

In the World Series the American League team (A) and the National League team (N) play until one team wins four games. If the sequence of winners is designated by letters (for example, NAAAA means that the National League team won the first game and the American League won the next four), how many different sequences are possible?

Letter Permutations How many different permutations of the letters in the word $CINCINNATI$ are there?

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?

The locks on the gymnasium lockers have dials with numbers from 0 to 39 . Each locker combination consists of 3 numbers.

a. Pythagoras remembers that his numbers are 8,15 , and 17 , but he can't remember in which order they appear. If he can try one possibility every 10 s, what is the maximum amount of time that it would take him to find the right combination?

b. Hypatia cannot remember any of her combination numbers, but she does remember that exactly two of them are the same. If she can try one possibility every 10 s, what is the maximum amount of time that it would take her to find the right combination?

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

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

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?

How many different 9-digit numbers can be formed by using the digits in the number 277,728,788?

a. In how many ways can the letters in the word XRAY be arranged? b. In how many ways can the letters in the word MAMMOGRAM be arranged?

Find the number $m$ of ways a judge can award first, second, and third places in a contest with $18$ contestants.