A committee of fifty politicians is to be chosen from among our one hundred U.S. senators. If the selection is done at random, what is the probability that each state will be represented?

Model the graph shown using a sine function. a) From the graph, determine the amplitude, the period, the phase shift, and the vertical translation. b) Write an equation for the function. c) Graph the equation of the function. d) Compare the graph in part c) to the given graph, and verify that the graphs match.

Use combinatorial reasoning to prove the mentioned identity.

A diode has $w_{d o}=0.4 \mu \mathrm{m}$ and $\phi_{j}=0.85 V$. (a) What reverse bias is required to triple the depletion-layer width? (b) What is the depletion region width if a reverse bias of 7 V is applied to the diode?

In how many ways can a pack of fifty-two cards be dealt to thirteen players, four to each, so that every player has one card of each suit?

How many ways can the letters in the word S L U M G U L L I O N be arranged so that the three L's precede all the other consonants?

a. In how many ways can the letters in the word FLORIDA be arranged? b. In how many ways can the letters in the word MISSISSIPPI be arranged?

Mitch is trying to add a little zing to his cabaret act by telling four jokes at the beginning of each show. His current engagement is booked to run four months. If he gives one performance a night and never wants to repeat the same set of jokes on any two nights, what is the minimum number of jokes he needs in his repertoire?

A new horror movie, Friday the 13th, Part X, will star Jason’s great-grandson (also named Jason) as a psychotic trying to dispatch (as gruesomely as possible) eight camp counselors, four men and four women. (a) How many scenarios (i.e., victim orders) can the screenwriters devise, assuming they want Jason to do away with all the men before going after any of the women? (b) How many scripts are possible if the only restriction imposed on Jason is that he save Muffy for last?

An arithmetic progression has first term a and common difference 10. The sum of the first n terms of the progression is 10 000. Express a in terms of n, and show that the nth term of the progression is $\frac{10000}{n}+5(n-1)$. Given that the nth term is less than 500, show that $n^{2}-101 n+2000<0$ and hence find the largest possible value of n.

Four men and four women are to be seated in a row of chairs numbered 1 through 8. (a) How many total arrangements are possible? (b) How many arrangements are possible if the men are required to sit in alternate chairs?

Suppose that ten people, including you and a friend, line up for a group picture. How many ways can the photographer rearrange the line if she wants to keep exactly three people between you and your friend?

How many ways can a set of four tires be put on a car if all the tires are interchangeable? How many ways are possible if two of the four are snow tires?

Compare the coefficients of $t^{k}$ in

$$(1+t)^{d}(1+t)^{e}=(1+t)^{d+e}$$

to prove that

$$\sum_{j=0}^{k}\left(\begin{array}{c} d \\ j \end{array}\right)\left(\begin{array}{c} e \\ k-j \end{array}\right)=\left(\begin{array}{c} d+e \\ k \end{array}\right)$$