450 CHAPTER 7 Counting and Combinatorics
lottery is based on trying to guess which six randomly picked numbers from the set
{1, 2 ... , 38} will be chosen. Winning a third state lottery is based on trying to guess7 of 11 randomly picked numbers from the set of {1, 2,..., 80). How many possible
winning combinations are there for each of these lotteries?- Given 1, 2 .... 11, select a subset of five elements from this set and a second subset
with two of these elements. In how many ways can these groups be formed if:
(a) There are no restrictions.
(b) Each group contains all even or all odd integers.
(c) No repetitions are allowed, and the smallest member of the second group is larger
than the largest member of the first group. Show that it does not matter whether
the two-element set or the five-element set is chosen first. - How many ways can a committee of three men and two women be chosen from six
men and four women? What if Adam Smith and Abigail Smith will not serve on the
same committee? - How many ways can a committee of three be chosen from four teams of two with each
team consisting of a man and a woman if:
(a) All are equally eligible.
(b) The committee must consist of two women and one man.
(c) A man and a woman from the same team cannot serve on the committee. - Two committees of five persons each must be chosen from a group of 375 people. If
the committees must be disjoint, in how many ways can the committees be chosen? If
the committees need not be disjoint, in how many ways can this be done? - A series of articles about 26 athletic teams will appear over 26 consecutive weeks. How
many ways can the articles be ordered? What if the first week's article must be about
the current champion? What if team A and team B must be featured in consecutive
weeks? - Solve the problem in Example 10 if the claim is that a different pizza can be ordered
every day for three years. - Magic Ice sells ice cream sundaes. With m flavors of ice cream and k possible toppings,
how large should m and k be so that a customer can have a different sundae every
night for five years? A sundae has just one kind of ice cream and at most three types
of toppings. - How many ways are there for a person to travel from the southwest corner to the
northeast comer of an m x n grid? Enumerate all the ways possible if the grid is 5 x 3.
How many ways are there if the grid is 10 x 10 and no move may take the person
below the main diagonal (those positions that are k steps over and k steps up from the
starting point where 1 < k < 10). - Five students (A, B, C, D, and E) are scheduled to present papers in class.
(a) How many ways can this be arranged?
(b) How many ways can this be arranged without B speaking before A?
(c) How many ways can this be arranged if A speaks immediately before B? - How many ways can 12 black pawns be placed on the black squares of an 8 x 8 chess
board? How many ways can 12 black pawns and 12 white pawns be placed on the
black squares of an 8 x 8 chess board? Half the 64 squares are black and half are red.
No black (red) square shares an edge with a black (red) square.