CHAPTER 1:
Basic Principle of Counting – r experiments to be performed, each with n possible outcomes, total of n
1
x n
2
x…x n
r
total outcomes.
General formula for permutations (order matters so abc ≠ bca: Given n objects, there are n! different ways to arrange them.
There are (n1)! Ways to arrange the n elements in a circle.
!
!
!
!
2
1
r
n
n
n
n
different permutations of n objects, of which n1 are alike, n2 are alike, etc.
!
)!
(
!
r
r
n
n
r
n

=
=
!
)
(
r
n
r
is the number of possible combinations of size r that could be made from n objects when order is irrelevant. The numerator of the
last term is just the notation for n PERMUTE r. Each expression is known as a binomial coefficient.

+


=
r
n
r
n
r
n
1
1
1
Binomial Theorem:
∑
=

=
+
n
k
k
n
k
n
y
x
k
n
y
x
0
)
(
If there are n distinct objects that we want to divide into r different groups of size n1,n2,…nr, we denote it:
!
!
!
!
2
1
,...
2
1
r
r
n
n
n
n
n
n
n
n
=
Balls into boxes equations (or elevators):
0^0^0^0^…^0, n objects denoted by 0, and choose r1 ways to put in a barrier ^


1
1
r
n
distinct positive integervalued vectors (x1,x2,..xr) satisfying x1+x2…+xr = n, x>0
If each xr just needs to be nonnegative (urns can be empty, not every floor needs someone to get off at it) then the equation goes to:


+
1
1
r
r
n
Chapter 1 Example Problems:
College planning committee of 3 freshmen, 4 sophomores, 5 juniors and 2 seniors. How many subcommittees are possible if 1 person is chosen from each
class?
Answer: 3*4*5*2 = 120.
Similar problems: Total of different ways to order dinner at a restaurant givern appetizers, entrees, desserts and drinks.
How many 7place license plates with first 3 as letters and last 4 as numbers?
Answer: 26*26*26*10*10*10*10 = 175,760,000.
What if no replacement is allowed on license plate problem?
Answer: 26 *25*24*10*9*8*7 = 78,624,000
Similar problems: Possible phone numbers
How many different batting orders possible for 9 players?
Answer: 9!
Have 4 math books, 3 chem, 2 history and 1 language. If you want to keep same subjects together, how many ways are there to order the books on the shelf?
Answer: 4! Ways to order math books * 3! Chem. * 2! History *1! Language, but there are also 4! To arrange the subjects before arranging the books within
the subject, so total ways is 4!4!3!2!1!=6912
How many different ways can you arrange the letters P E P PE R?
Answer: 6! Ways to arrange 6 letters, but with 3 P’s and 2 E’s, many of the
arrangements will look the same, so divide by different ways to arrange those letters. 6!/(3!2!) = 60
Given 5 women and 7 men, how many committees of 2 women and 3 men can be formed?
Answer: (5 choose 2)*(7 choose 3) = 350.
What if two of the
men are fighting?
Answer: (2 choose 2)*(5 choose 1) = 5 of the subgroups of men contain the fighting guys, so 30 of the subgroups of men don’t. 30*(5 choose 2)
subgroups of women = 300 groups.