91Ó°ÊÓ

For each of the four subsets of the two properties (a) and (b), count the number of four-digit numbers whose digits are either \(1,2,3,4\), or \(5:\) (a) The digits are distinct. (b) The number is even. Note that there are four problems here: \(\emptyset\) (no further restriction), \\{a\\} (property (a) holds), \\{b\\} (property (b) holds), \(\\{a, b\\}\) (both properties (a) and (b) hold).

How many orderings are there for a deck of 52 cards if all the cards of the same suit are together?

Determine the largest power of 10 that is a factor of the following numbers (equivalently, the number of terminal 0s, using ordinary base 10 representation): (a) \(50 !\) (b) \(1000 !\)

How many integers greater than 5400 have both of the following properties? (a) The digits are distinct. (b) The digits 2 and 7 do not occur.

In how many ways can four men and eight women be seated at a round table if there are to be two women between consecutive men around the table?

In how many ways can six men and six women be seated at a round table if the men and women are to sit in alternate seats?

How many sets of three integers between 1 and 20 are possible if no two consecutive integers are to be in a set?

A football team of 11 players is to be selected from a set of 15 players, 5 of whom can play only in the backfield, 8 of whom can play only on the line, and 2 of whom can play either in the backfield or on the line. Assuming a football team has 7 men on the line and 4 men in the backfield, determine the number of football teams possible.

A classroom has two rows of eight seats each. There are 14 students, 5 of whom always sit in the front row and 4 of whom always sit in the back row. In how many ways can the students be seated?

In how many ways can six indistinguishable rooks be placed on a 6 -by-6 board so that no two rooks can attack one another? In how many ways if there are two red and four blue rooks?