A Counting Tutorial - Part 3

This tutorial uses strategies and counting methods presented in the previous tutorials. A problem is solved to illustrate some of the previous the methods. The tutorial closes with a list of problems the reader is encouraged to try in order to master the techniques previously demonstrated.

To emphasis the importance of the counting principles, the counting rules are cited in the solution. Let's begin!

Problem: In how many different ways can eight people be seated around a round table?

In a previous tutorial, we learned how to count the number of arrangements of eight distinguishable objects (in a row). This seems like a good place to start. So the quick response would be 8!. But it's not correct. The 8!-count implies all of the arrangements are different. This is not true for circular seating!

Let's see why.

Designate the people to be seated as a, b, c, d, e, f, and g. Further, assume the first position in an arrangement is the "head chair". The 8!- count gives the following arrangements as different.

a, b, c, d, e, f, g, h b, c, d, e, f, g, h, a
c, d, e, f, g, h, a, b d, e, f, g, h, a, b, c
e, f, g, h, a, b, c, d f, g, h, a, b, c, d, e
g, h, a, b, c, d, e, f h, a, b, c, d, e, f, g

In the above arrangements, each person is an occupant in the head chair. Also for seating around a round table, these seat assignments place the sitters in the same relative position to each other, albeit in different seats.

Verify: Place two of these arrangements around a circle.

These arrangements are called cyclic arrangements because an arrangement immediately following the proceeding arrangement is generated when the head chair occupant of the preceding arrangement is moved to the end of the list.

Observe also that an arrangement has the same number of cyclic arrangements, namely 8,

Now let's count the different seating arrangements at a round table?

Let N be the number of different seating arrangements of eight people seated on a round table. To simply the explanation, call these arrangements original.

List the original N arrangements in a column.
Along each row of an original, list all of its cyclic arrangements. There are seven others.

This procedure lists all arrangements of 8 distinguishable letters in a row in a table having N rows and 8 columns.

Hence 8 N = 8! , therefore

N=8!/8 = 7! = 5040.

More generally, the total number of arrangements of n distinguishable objects around a circle is (n-1)! = n!/n.

Remark: The product rule asserts that the number of arrangements in the table is 8 N.

The other counting tutorials:
A_Counting_Tutorial
A_Counting_Tutorial - Part_2

To practice counting rules try there:
Try these:
1. Find the number of ways to arrange the letters in the words success, algebraic, Uranus, Mississippi.
2. In how many ways can the letters aaabbbccc be arranged in a row?
3. How many different numbers can be formed using the digits 1,2,3,4,5,6? No repetition.
4. In how many ways can 10 boys be placed in a row so that two particular boys are not together?
5. How many numbers can be formed from the digits 1,2,3,4,5 that are greater than 23,000? Repetition is not allowed?