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In how many ways can the 11 identical horses on a carousel be painted so that three are brown, three are white, and five are black?

In how many ways can a teacher distribute 12 different science books among 16 students if (a) no student gets more than one book? (b) the oldest student gets two books but no other student gets more than one book?

Four numbers are selected from the following list of numbers: \(-5,-4,-3,-2,-1,1,2,3,4\). (a) In how many ways can the selections be made so that the product of the four numbers is positive and (i) the numbers are distinct? (ii) each number may be selected as many as four times? (iii) each number may be selected at most three times? (b) Answer part (a) with the product of the four numbers negative.

a) How many permutations are there for the eight letters \(\mathrm{a}, \mathrm{c}, \mathrm{f}, \mathrm{g}, \mathrm{i}, \mathrm{t}, \mathrm{w}, \mathrm{x} ?\) b) Consider the permutations in part (a). How many start with the letter t? How many start with the letter \(\mathrm{t}\) and end with the letter \(\mathrm{c}\) ?

In how many ways can we distribute eight identical white balls into four distinct containers so that (a) no container is left empty? (b) the fourth container has an odd number of balls in it?

Waterbury Hall, a university residence hall for men, is operated under the supervision of Mr. Kelly. The residence has three floors, each of which is divided into four sections. This coming fall Mr. Kelly will have 12 resident assistants (one for each of the 12 sections). Among these 12 assistants are the four senior assistants -Mr. DiRocco, Mr. Fairbanks, Mr. Hyland, and Mr. Thornhill. (The other eight assistants will be new this fall and are designated as junior assistants.) In how many ways can Mr. Kelly assign his 12 assistants if a) there are no restrictions? b) Mr. DiRocco and Mr. Fairbanks must both be assigned to the first floor? c) Mr. Hyland and \(\mathrm{Mr}\). Thomhill must be assigned to different floors?

a) Fifteen points, no three of which are collinear, are given on a plane. How many lines do they determine? b) Twenty-five points, no four of which are coplanar, are given in space. How many triangles do they determine? How many planes? How many tetrahedra (pyramidlike solids with four triangular faces)?

How many ways are there to place 12 marbles of the same size in five distinct jars if (a) the marbles are all black? (b) each marble is a different color?

In the Internet each network interface of a computer is assigned one, or more, Internet addresses. The nature of these Internet addresses is dependent on network size. For the Internet Standard regarding reserved network numbers (STD 2), each address is a 32 -bit string which falls into one of the following three classes: (1) A class A address, used for the largest networks, begins with a 0 which is then followed by a seven-bit network number, and then a 24-bit local address. However, one is restricted from using the network numbers of all 0 's or all 1's and the local addresses of all 0 's or all 1's. (2) The class B address is meant for an intermediate-sized network. This address starts with the two-bit string 10, which is followed by a 14-bit network number and then a 16 -bit local address. But the local addresses of all 0 's or all 1's are not permitted. (3) Class C addresses are used for the smallest networks. These addresses consist of the three-bit string 110 , followed by a 21 -bit network number, and then an eight-bit local address. Once again the local addresses of all 0 's or all 1's are excluded. How many different addresses of each class are available on the Internet, for this Internet Standard?

A computer science professor has seven different programming books on a bookshelf. Three of the books deal with \(\mathrm{C}++\), the other four with Java. In how many ways can the professor arrange these books on the shelf (a) if there are no restrictions? (b) if the languages should alternate? (c) if all the C++ books must be next to each other? (d) if all the C++ books must be next to each other and all the Java books must be next to each other?