COMBINATORICS
Example:
A senator must hire a new chief-of-staff and a new press secretary.
If there are 5
applicants for chief-of-staff and 3 applicants for press secretary, how many different ways can
these two positions be filled?
Solution:
Let’s call the applicants for chief-of-staff A, B, C, D, and E.
We’ll name the
applicants for press secretary X, Y, and Z.
The following
tree diagram
shows all the
possibilities:
We see that there are 15 different ways to make the two hires.
Do you see a shortcut we could have used to get the answer without drawing the tree
diagram?
Fundamental Principle of Counting: If the first choice can be made in x ways and the second
choice can be made in y ways, then the choices can be made consecutively in xy ways.
This can be generalized to situations where more than two consecutive choices are being made.
Example:
All of a sudden, the senator’s director of legislative affairs quits.
If there are two
applicants for this position, in how many ways can the senator fill the three positions?
Solution:
Using the Fundamental Principle of Counting, (5)(3)(2) = 30 ways.
Example:
A voter has to rank 5 alternatives.
How many different preference lists are possible?
Solution:
(5)(4)(3)(2)(1) = 120
Products that start with a number and multiply by every integer down to one are very common in
mathematics and we have a special notation for them.
(5)(4)(3)(2)(1) = 5! (pronounced five
factorial
)
In general:
n! = n(n-1)(n-2)
....
(3)(2)(1) if n is a positive integer
0! = 1

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