MATH 351 Fall 2009 Homework 1 Due: Wednesday, September 30

( ) 60
To count this, first choose the 30 men for dorm A from the 60 possible men by .
(
30
) 70
Second choose the 35 women from the 70 women, that will go into dorm B by .
35
The remainder will then be in dorm C. By the basic principle of counting the total
possible placings of the students is the product:
( ) ( ) 60 70 60!70!
=
30 35 30!30!35!35! .
Problem 3. In how many ways can ten people be seated in a row if:
(a) there are no restrictions on the seating arrangement?
This is a simple permutation of 10 distinct objects. Thus the solution is 10! =.
(b) persons A and B must sit next to each other?
If A and B must sit together we treat them as one ”person” and order them with the
remaining 8 people in 9! different ways. Then for each of those orderings we either
have A first or B first or 2 possible arrangements. By the basic counting principle we
have a total of 2 × 9! = orderings.
(c) there are 5 men and 5 women, and no 2 men or 2 women can sit next to each
other?
First choose if they sit mwmwmwmwmw or wmwmwmwmwm. Once this is chosen
then order the men in their seats. Since there are five of them there are 5! arrangements.
Similarly the women can then be seated in 5! different ways. Thus the total
number of arrangements is 2 × 5! × 5! =.
(d) there are 5 men and they must sit next to each other?
2