Name:
Date:
4.2
Introducing Permutations
and Factorial Notation
YOU WILL NEED
• calculator
Keep in Mind
The number of permutations, or ordered arrangements, of a set of n different
objects is given by the expression n! 5 n(n 2 1)(n 2 2)…(3)(2)(1), called
n factorial. In this context, the expression is defined for natural numbers.
Example 1
A six-digit secret code number uses the digits 1 through 6 exactly once each. If the
first digit of the code number is 3 and the last digit is 2, how many possibilities are
there for the code number?
Solution
I knew that the first digit was 3 and the last
3
2
digit was 2, so the middle four digits of the
code number could be any permutation of the set {1, 4, 5, 6}. There are 4 different
digits in this set, so the number of permutations is 4! 5 (4)(3)(2)(1), or 24. There are
24 possibilities.
Example 2
a) Simplify
(n 1 3)(n 1 2)!
.
(n 1 1)!
(n 1 3)(n 1 2)!
5 30
(n 1 1)!
where n [ N.
b) Solve
Solution
a) I wrote each factorial as a product and then looked for common factors in the
numerator and denominator.
(n 1 3) # (n 1 2)!
(n 1 3)(n 1 2)(n 1 1)(n) c(3)(2)(1)
5
(n 1 1)!
(n 1 1)(n) c(3)(2)(1)
5
(n 1 3)(n 1 2)(n 1 1)!
(n 1 1)!
5 (n 1 3)(n 1 2)
5 n2 1 5n 1 6
94
4.2
Introducing Permutations and Factorial Notation
FoM12 WB_Ch04W_BLM.indd 94
NEL
7/12/12 12:05 PM
Name:
Date:
b) I used the simplified expression from part a) to write a quadratic equation. I then
TIP
solved this equation by factoring.
When n 5 28, the
expressions
(n 1 2)! 5 (26)!
and (n 1 1)! 5 (27)! are
undefined. So, n 5 28
could not be a solution in
any case.
n2 1 5n 1 6 5 30
n2 1 5n 2 24 5 0
(n 1 8)(n 2 3) 5 0
n1850
n2350
or
n 5 28
n53
I knew that n must be a natural number, so the solution is n 5 3.
Practice
1. a) How many arrangements are possible using all of the letters in WHISTLER?
b) How many arrangements of the letters in WHISTLER end with a W?
2. Evaluate the following expressions.
a) 5! 5
b) 7 ? 6! 5
c)
6!
5
3!
d)
7!
5
3!2!
c)
(n 1 4)!
(n 1 2)!
d)
(n 2 1)!
(n 1 1)!
3. Simplify the following expressions, where n [ I.
a)
(n 1 1)!
n!
b) n(n 2 1)(n 2 2)!
NUMERICAL RESPONSE
4. Solve the equation
n!
5 20, where n [ I.
(n 2 2)(n 2 3)!
n(n 2 1)(
n!
5
(n 2 2)(n 2 3)!
(
Check each possible
solution for n to decide
whether all the factorials
are defined.
)!
)!
) 5 20
n(
So,
TIP
2
2
5 20
50
(
)(
n2
)50
50
n5
NEL
FoM12 WB_Ch04W_BLM.indd 95
or
n1
50
n5
or
4.2
Introducing Permutations and Factorial Notation
95
7/12/12 12:06 PM