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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?

During a local campaign, eight Republican and five Democratic candidates are nominated for president of the school board. a) If the president is to be one of these candidates, how many possibilities are there for the eventual winner? b) How many possibilities exist for a pair of candidates (one from each party) to oppose each other for the eventual election? c) Which counting principle is used in part (a)? in part (b)?

Calculate \(\left(\begin{array}{l}6 \\ 2\end{array}\right)\) and check your answer by listing all the selections of size 2 that can be made from the letters a, b, c, d, e, and \(\mathrm{f}\).

Twelve points are placed on the circumference of a circle and all the chords connecting these points are drawn. What is the largest number of points of intersection for these chords?

a) In how many ways can one travel in the \(x y\)-plane from \((0,0)\) to \((3,3)\) using the moves \(\mathrm{R}:(x, y) \rightarrow(x+1, y)\) and \(\mathrm{U}:(x, y) \rightarrow(x, y+1)\), if the path taken may touch but never fall below the line \(y=x\) ? In how many ways from \((0,0)\) to \((4,4) ?\) b) Generalize the results in part (a). c) What can one say about the first and last moves of the paths in parts (a) and (b)?

Buick automobiles come in four models, 12 colors, three engine sizes, and two transmission types. (a) How many distinct Buicks can be manufactured? (b) If one of the available colors is blue, how many different blue Buicks can be manufactured?

A choir director must select six hymns for a Sunday church. service. She has three hymn books, each containing 25 hymns (there are 75 different hymns in all). In how many ways can she select the hymns if she wishes to select (a) two hymns from each book? (b) at least one hymn from each book?

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?

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