The following exercise is a problem that may require permutations, combinations, or the multiplication principle.

According to the Baskin-Robbins Web site, there are 21 "classic flavors" of ice cream.

(a) How many different double-scoop cones can be made if order does not matter (for example, putting chocolate on top of vanilla is equivalent to putting vanilla on top of chocolate)?

(b) How many different triple-scoop cones can be made if order does matter?

Washtenaw Dairy in Ann Arbor, Michigan, sells 63 flavors of Strohs Mooney’s ice cream, and Ben and Jerry’s Web site also lists 63 different flavors of ice cream. These numbers are similar to the varieties of smartphones sold by Samsung, Nokia, and Motorola. Toyota makes only 16 varieties of vehicles and Boeing makes 16 varieties of airplanes. Why is there more variety in smartphones and ice cream than in automobiles and airplanes?

Garcia Co. had three major business transactions during 2014.

(c) Garcia Co. wanted to make its 2014 income look better, so it added 2 more weeks to the year (a 54-week year). Previous years were 52 weeks.

Instructions

In each situation, identify the assumption or principle that has been violated, if any, and discuss what the company should have done.

When 2.00 mo! of $SO_2Cl_2$ is placed in a 2.00-L fla sk at 303 K, 56% of the $SO_2Cl_2$ decomposes to $SO_2$ and $Cl_2:$ $\mathrm { SO } _ { 2 } \mathrm { Cl } _ { 2 } ( g ) \rightleftharpoons \mathrm { SO } _ { 2 } ( g ) + \mathrm { Cl } _ { 2 } ( g )$ According to Le Chatelier's principle, would the percent of $SO_2Cl_2$ that decomposes increase, decrease or stay the same if the mixture were transferred to a 15.00-L vessel?

What is the expected result when you roll a single die? Specify 40 rolls, and then click on "roll the dice." After the last roll, click on "show mean." Does your expected sum agree with the green mean? What happens to the plotted points as the number of rolls increases? If the picture is not clear, click on "roll the dice" for a new total of 80 rolls. Describe what you see in a sentence. What is the name for this principle?

The crate of mass m is pushed to the left until the linear elastic spring of constant k has compressed a distance d from its unstretched length $\ell_{0}$. The crate is released from rest, and friction between the crate and the horizontal surface is negligible.Using the work-energy principle, determine the value of k so that the crate is moving at 3 m/s the instant the spring becomes uncompressed. Use m = 20 kg and d = 0.75 m.

The owner of Bun ‘N’ Run Hamburgers wishes to compare the sales per day at two locations. The mean number sold for 10 randomly selected days at the Northside site was 83.55, and the standard deviation was 10.50. For a random sample of 12 days at the Southside location, the mean number sold was 78.80 and the standard deviation was 14.25. At the .05 significance level, is there a difference in the mean number of hamburgers sold at the two locations? What is the p-value?

The length of time (in seconds) that a user views a page on a Web site before moving to another page is a lognormal random variable with parameters $\theta=0.5 \text { and } \omega^{2}=1$ (a) What is the probability that a page is viewed for more than 10 seconds? (b) What is the length of time that 50% of users view the page? (c) What is the mean and standard deviation of the time until a user moves from the page?

Search the Internet for the multiple listings in your area. If you cannot gain access without a subscription, go to the web site of a large real estate agency that operates in your area. Create a bulleted list of the types of information supplied in the house listings. Look at the listings for several houses in your neighborhood. Write a brief summary of the kinds of houses available near you, including their price range, house styles, internal features, and acreage.

Refer to the figure and use the principle of superposition to Find the component of the current $i$ through $R_{3}$ that is due to $V_{S 2}$.

$$\begin{aligned} & V_{S 1}=V_{S 2}=450 \mathrm{~V} \\ & R_{1}=7 \Omega \quad R_{2}=5 \Omega \\ & R_{3}=10 \Omega \quad R_{4}=R_{5}=1 \Omega \end{aligned}$$

Suppose you attempt to pick up a very heavy object. Before you tried to pick it up, the object was sitting still—its momentum was not changing. You pull very hard, but do not succeed in moving the object. Is this a violation of the Momentum Principle? How can you be exerting a large force on the object without causing a change in its momentum? What does change when you apply this force?

In a population of rabbits, f (C$_{1}$) = $0.70$ and f (C$_{2}$) = $0.30$. The alleles exhibit an incomplete dominance relationship in which C$_{1}$C$_{1}$ produces black rabbits, C$_{1}$C$_{2}$ tan-colored rabbits, and C$_{2}$C$_{2}$ rabbits with white fur. If the assumptions of the Hardy-Weinberg principle apply to the rabbit population, what are the expected frequencies of black, tan, and white rabbits?

For the following reports about statistical studies, identify the following items (if possible). If you can't tell, then say so-this often happens when we read about a survey.

The sample

A question posted on the gamefaqs.com website on August 1, 2011, asked visitors to the site, "Do you have an active social life outside the Internet?" 22% of the 55,581 respondents said "No" or "Not really, most of my personal contact is online."

The lines containing the altitudes of a triangle meet at a single point, called the orthocenter. The position of the orthocenter is determined by the measures of the angles. Construct the three altitudes for each of the following triangles. Then complete the statements in parts a-c.

Use the Chapter $14$ Geometer's Sketchpad activity Orthocenter on our Web site to investigate a wide range of acute, obtuse, and right triangles. Do your conclusions for parts $\mathrm{a}, \mathrm{b}$, and $c$ still hold?