A Counting Tutorial - Part 3 Corrected

This tutorial uses strategies and counting methods presented in previous tutorials. To illustrate previously discussed methods, a problem is solved. The importance of the counting principles is emphasized by citing it in the solution of the problem. The tutorial closes with a list of problems to encourage the reader to practice the counting techniques.

Let's begin!

Problem: 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 that all 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, g and h. The 8!- count asserts 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 list,

• The first position in the arrangement is a "head chair".
• Each of the eight people is an occupant in the head chair in one arrangement.

For seating around a round table, these seat assignments do not change the relative seating position of the eight people. They only change the chair in which they sit.

Verify: Place two of these arrangements around a circle.

Call these arrangements cyclic. For each is followed by the arrangement where the head chair occupant has moved to the end of the arrangement.

For example the arrangement a, b, c, d, e, f, g, h becomes b, c, d, e, f, g, h, a when the head chair occupant moves to the end of the list.

Note all arrangements have exactly eight cyclic arrangements,

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 at a round table. To simply the explanation, call each of these arrangements an original arrangement.

List the original N original arrangements in a column.

Along each row headed by an original arrangement, list all its cyclic arrangements. There will be seven.

This listing procedure enumerates into a table all of the arrangements of the 8 distinguishable letters.

There are 8! such arrangements.

Since the table has N rows and 8 columns. 8 N = 8! . Thus 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 elements in the table is 8 N.

Practice counting rules by supplying answers to the following:
1. Find the number of ways to arrange the letters in the words: success, algebraic, Uranus, and Mississippi.

2. In how many ways can the letters a a a b b b c c c be arranged in a row?

3. How many different numbers can be formed using all of 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, greater than 23,000, can be formed from the digits 1, 2, 3, 4, 5 ? Repetition is not allowed?