The expression 6 × 5 × 4 × 3 × 2 × 1 = can be
written as 6!, which is read as “six factorial.”
In general, n! is the product of all the counting
numbers beginning with n and counting backwards
We define 0! to be 1.
Factorial on your TI calculator.
Find the value of each expression:
c) 3! + 2!
Fundamental Counting Principle:
If one activity can occur in any of m ways and,
following this, a second activity can occur in any
of n ways, then both activities can occur in the
order given in m*n ways.
Permutation Formula an arrangement of objects in some specific order
In general P(n, r) means the number of permutations
of n items arranged r at a time.
The formula for permutation is
Permutations on your TI calculator
nPn = n!
3P3 = 3! = 3*2*1 = 6
Words used in permutation problems:
• line up
• president, vice president, secretary
• 1st, 2nd, 3rd place
A license plate begins with three letters. If the possible letters are A, B, C, D and
E, how many different arrangements of these letters can be made if no letter is
used more than once?
Permutations with repetition
If we want to arrange items when there are more than
one of the same item, we need to divide by the number
of identical items:
Find the number of arrangements of the letters that can
be formed from the letters IDENTITY, using each letter
Example: Find the number of arrangements of
letters that can be formed from the letters:
An arrangement of objects in which the order is
not important is called a combination. This is
different from permutation where the order
matters. For example, suppose we are arranging
the letters A, B and C. In a permutation, the
arrangement ABC and ACB are different. But, in a
combination, the arrangements ABC and ACB are
the same because the order is not important.
The number of combinations of n things taken r at
a time is written as C(n, r).
The formula is given by:
Combinations on your TI calculator
Words used in combination problems:
In how many ways can a coach choose three
swimmers from among five swimmers?
There are 5 red and 4 white marbles in an urn. A marble is drawn from the urn and not
replaced. Then, a second marble is drawn.
a. In how many ways can a red marble and a white marble be drawn in that order?
b. In how many ways can a red marble and a white marble be drawn in either order?
An urn contains three white balls and four red balls. Two balls are chosen at
random. How many ways can you chose at least one of the red balls?