1. Chapter 10
Counting Techniques

2. Miscellaneous Counting
Methods Section 10.4

3. Permutations and Combinations
Three cards are dealt in succession and without
replacement from a standard deck of 52 cards.
In how many different orders can the
cards be dealt?
How many different 3-card hands are possible?
52 nPr 3 = 52 x 51 x 50 = 132,600
52 x 51 x 50
3 x 2 x 1
52 nCr 3 =
= 22,100

4. Permutations and Combinations
The playbook for the quarterback of the Dallas Cowboys
contains 50 plays.
In how many different ways could the
quarterback select 3 plays to use in
succession in the next three downs?
In how many different ways could he
select a set of 3 plays to study?
50 x 50 x 50 = 125,000
50 x 49 x 48
3 x 2 x 1
50 nCr 3 =
= 19,600

5. Permutations and Combinations
Peter must select three electives from a group of 7 courses.
In how many ways can Peter do this?
If all 7 courses are available each of the 4 hours from 8 AM to noon, from how many
different schedules (hours and what course
at each hour) can Peter choose?
7 x 6 x 5
3 x 2 x 1
7 nCr 3 =
= 35
7 nPr 4 = 7 x 6 x 5 x 4= 840

6. Distinct Arrangements
The total number of distinct arrangements of n objects, where p objects are identical, q objects are identical, r objects are identical, and so on, is given by

7. Distinct Arrangements
How many different arrangements can be made
with the letters in the word TALLAHASSEE?
with the letters in the word MISSISSIPPI?
11!
3!2!2!2!
39,916,800
6x2x2x2
39,916,800 48 = = =
831,600
11!
4!4!2!
39,916,800
24x24x2
39,916,800
1152 = = =
34,650
END