91Ó°ÊÓ

We are given eight, rooks, five of which are red and three of which are blue. (a) In how many ways can the eight rooks be placed on an 8 -by- 8 chessboard so that no two rooks can attack one another? (b) In how many ways can the eight rooks be placed on a 12 -by-12 chessboard so that no two rooks can attack one another?

Determine the number of circular permutations of \(\\{0,1,2, \ldots, 9\\}\) in which 0 and 9 are not opposite. (Hint: Count those in which 0 and 9 are opposite.)

How many permutations are there of the letters of the word ADDRESSES? How many 8-permutations are there of these nine letters?

A footrace takes place among four runners. If ties are allowed (even all four runners finishing at the same time), how many ways are there for the race to finish?

We are to seat five boys, five girls, and one parent in a circular arrangement around a table. In how many ways can this be done if no boy is to sit next to a boy and no girl is to sit next to a girl? What if there are two parents?

In a soccer tournament of 15 teams, the top three teams are awarded gold, silver, and bronze cups, and the last three teams are dropped to a lower league. We regard two outcomes of the tournament as the same if the teams that receive the gold, silver, and bronze cups, respectively, are identical and the teams which drop to a lower league are also identical. How many different possible outcomes are there for the tournament?

Determine the number of 11 -permutations of the multiset $$ S=\\{3 \cdot a, 4 \cdot b, 5 \cdot c\\} . $$

Determine the number of 10-permutations of the multiset $$ S=\\{3 \cdot a, 4 \cdot b, 5 \cdot c\\} . $$

List all 3-combinations and 4-combinations of the multiset $$ \\{2 \cdot a, 1 \cdot b, 3 \cdot c\\} $$

Determine the total number of combinations (of any size) of a multiset of objects of \(k\) different types with finite repetition numbers \(n_{1}, n_{2}, \ldots, n_{k}\), respectively.