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In how many ways can 10 (identical) dimes be distributed among five children if (a) there are no restrictions? (b) each child gets at least one dime? (c) the oldest child gets at least two dimes?

A machine has nine different dials, each with five settings labeled \(0,1,2,3\), and 4 . a) In how many ways can all the dials on the machine be set? b) If the nine dials are arranged in a line at the top of the machine, how many of the machine settings have no two adjacent dials with the same setting? c) How many machine settings in part (b) use only 0,2 , and 4 as dial settings?

In how many ways can 15 (identical) candy bars be distributed among five children so that the youngest gets only one or two of them?

Evaluate each of the following. a) \(C(10,4)\) b) \(\left(\frac{12}{7}\right)\) c) \(C(14,12)\) d) \(\left(\begin{array}{l}15 \\ 10\end{array}\right)\)

A certain ice cream store has 31 flavors of ice cream available. In how many ways can we order a dozen ice cream cones if (a) we do not want the same flavor more than once? (b) a flavor may be ordered as many as 12 times? (c) a flavor may be ordered no more than 11 times?

a) The board of directors of a pharmaceutical corporation has 10 members. An upcoming stockholders' meeting is scheduled to approve a new slate of company officers (chosen from the 10 board members). How many different slates consisting of a president, vice president, secretary, and treasurer can the board present to the stockholders for their approval? b) Three members of the board of directors (from part a) are physicians. How many slates from part (a) have i) a physician nominated for the presidency? ii) exactly one physician appearing on the slate? iii) at least one physician appearing on the slate?

a) In how many ways can we select five coins from a collection of 10 consisting of one penny, one nickel, one dime, one quarter, one half-dollar, and five (identical) Susan B. Anthony dollars? b) In how many ways can we select \(n\) objects from a collection of size \(2 n\) that consists of \(n\) distinct and \(n\) identical objects?

While on a Saturday shopping spree Jennifer and Tiffany witnessed two men driving away from the front of a jewelry shop, just before a burglar alarm started to sound. Although everything happened rather quickly, when the two young ladies were questioned they were able to give the police the following information about the license plate (which consisted of two letters followed by four digits) on the getaway car. Tiffany was sure that the second letter on the plate was either an \(\mathrm{O}\) or a \(\mathrm{Q}\) and the last digit was either a 3 or an 8. Jennifer told the investigator that the first letter on the plate was either a \(\mathrm{C}\) or a \(G\) and that the first digit was definitely a 7 . How many different license plates will the police have to check out?

a) How many permutations of size 3 can one produce with the letters \(m, r, a, f\), and \(t\) ? b) List all the combinations of size 3 that result for the letters \(\mathrm{m}, \mathrm{r}, \mathrm{a}, \mathrm{f}\), and \(\mathrm{t}\).

To raise money for a new municipal pool, the chamber of commerce in a certain city sponsors a race. Each participant pays a \(\$ 5\) entrance fee and has a chance to win one of the different-sized trophies that are to be awarded to the first eight runners who finish. a) If 30 people enter the race, in how many ways will it be possible to award the trophies? b) If Roberta and Candice are two participants in the race, in how many ways can the trophies be awarded with these two runners among the top three?