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3.5 Combinations and Permutations
Recall from Section 3.2 the problem faced by a corn growing business: the FDA determines that
two of the 20 bushels are potentially contaminated with E. coli. Two bushels had been shipped
out and the question was: what is the probability that both bushels that were shipped to the local
grocer were uncontaminated
We wrote the simultaneous probability as
( )
( ) ( )
Due to the fact that one of the uncontaminated bushels was removed from the “pool”, there was
now only a 17/19 chance that the second uncontaminated bushel would be pulled. In short, we
wrote:
( )
We notice that the numerator and denominator both have a product of two sequential numbers.
Had they shipped, say, four bushels, the probability that all four were uncontaminated would be:
As you might imagine, this pattern continues.
How painful, though, would it be to have to multiply eight or nine probabilities of this nature
together You could certainly do it, but you might think, “It sure would be nice to take advantage
of this pattern!” Well, we‟re in luck!
Let‟s define an important term:
A factorial is a descending product of whole numbers down to 1, beginning at a specified whole
number. To start with a generic whole number, , we denote this product by , and write:
( ) ( )
Example 1: Find .
SOLUTION: By definition of factorial, we write
Statistics for Decision-Making in Business © Milos Podmanik Page 119