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. $(20-1)^3$

Draw a Venn diagram and shade the region representing the set given in part (a). Then draw a separate Venn diagram and shade the region representing the set given in part (b).

a. $P \cap Q$

b. $P \cup Q$

Find the value of expression.

a. ${ }_5 P_2$

b. ${ }_5 C_2$

In a parking lot containing 85 cars, there are 45 cars with automatic transmissions, 43 cars with rear-wheel drive, and 46 cars with four-cylinder engines. Of the cars with automatic transmissions, 26 also have rear-wheel drive. Of the cars with rear-wheel drive, 29 also have four-cylinder engines. Of the cars with four-cylinder engines, 27 also have automatic transmissions. There are 21 cars with all three features.

a. How many cars do not have automatic transmissions and rear-wheel drive but do have four-cylinder engines?

b. How many cars do not have any of the three features?

For each solution, calculate the initial and final pH after adding 0.010 mol of Na OH. a. 250.0 mL of pure water b. 250.0 mL of a buffer solution that is 0.195 M in $HCHO_2$ and 0.275 M in $KCHO_2$ c. 250.0 mL of a buffer solution that is 0.255 M in $CH_3CH_2NH_2$ and 0.235 M in $CH_3CH_2NH_3Cl$

The force F = 4i - 7j moves an object 4 ft along the x-axis in the positive direction. Find the work done if the unit of force is the pound.

Points P, Q, and R are given. a. Find the general equation of the plane passing through P, Q, and R. b. Write the vector equation $\mathbf { n } \cdot \vec { P S } = 0$ of the plane at a., where S(x, y, z) is an arbitrary point of the plane. c. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through P, Q, and R. P(1, 1, 1), Q(2, 4, 3), and R(-1, -2, -1)

Find an explicit formula for the sequence and use it to find the $12$th term.

$9,7.2,5.76,4.608,3.6864, \ldots$.

Let T be the linear transformation from R2 to R2 consisting of reflection in the y-axis. Let S be the linear transformation from R2 to R2 consisting of clockwise rotation of 30◦. (b) Find the standard matrix of T , [T ]. If you are not sure what this is, see p. 216 and more generally section 3.6 of your text. Do that before you go looking for help!

$A, B$, and $C$ denote $n\times n$ matrices. Assuming the law $\det (AB) = (\det A)(\det B)$, prove that $\det (ABC) = (\det A)(\det B)(\det C)$.

What are the two subsets of the real numbers that form the set of real numbers?

a) Define the union, intersection, difference, and symmetric difference of two sets. b) What are the union, intersection, difference, and symmetric difference of the set of positive integers and the set of odd integers?