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. $A \cap U=$ ?

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

Give the expansion of the binomial. Simplify your answer.

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

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?

Three identical door prizes are to be given to three lucky people in a crowd of 100. In how many ways can this be done?

Find the number of permutations of the letters.

MISSOURI

Evaluate:

a. 2!

b. 3 !

c. 4 !

Of the 320 children at a movie, 190 buy popcorn and 245 buy something to drink. If 151 children buy both, how many buy neither?

A certain small college has 1000 students. Let F= the set of college freshmen, and let M= the set of music majors. Describe the following set in words and tell how many members it has.

$$F \cap M$$

Find the number of permutations of the letters.

a. MALE

b. MALL

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

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?

Evaluate:

a. 5!

b. 6 !

c. 7 !

d. 0 !

Give the expansion of the binomial. Simplify your answer.

a. $(a+b)^3$

b. $(20+1)^3$

c. $(20-1)^3$