Permutation in Statistics

1. Review

2. Review Activity

3. Question 1
An English test contains five
different essay questions labeled A, B,
C, D, and E. You are supposed to
choose 2 to answer. How many
different ways are there to do this?

4. A voicemail system password is 1
letter followed by a 3-digit number
less than 600. How many different
voicemail passwords are possible if all
digits are allowed?
Question 2

5. A family of 3 plans to sit in the
same row at a movie theater. How
many ways can the family be seated
in 3 seats?
Question 3

6. Ingrid is stringing 3 different
types of beads on a bracelet. How
many ways can she string the next
three beads if they must include one
bead of each type?
Question 4

7. Nathan wants to order a
sandwich with two of the following
ingredients: mushroom, eggplant,
tomato, and avocado. How many
different sandwiches can Nathan
choose?
Question 5

8. A group of 8 swimmers are
swimming in a race. Prizes are
given for first, second, and third
place. How many different
outcomes can there be?
Question 6

9. How many different ways
can 9 people line up for a
picture?
Question 7

10. Four people need to be selected
from a class of 15 to help clean up
the campus. How many different
ways can the 4 people be chosen?
Question 8

11. Question 9
Four people need to be selected
from a class of 15 to help clean up
the campus. How many different
ways can the 4 people be chosen, if
the only two girls refuse to help?

12. A basketball team has 12
members who can play any position.
How many different ways can the
coach choose 5 starting players?
Question 10

13. A basketball team has 12
members who can play any
position. How many different ways
can the coach choose 5 starting
players if the captain MUST play the
first half?
Question 11

14. When ordering a pizza, you can
choose 2 toppings from the following:
mushrooms, olives, pepperoni,
pineapple, and sausage. How many
different types of pizza can you order?
Question 12

15. Nine people in a writing contest
are competing for first, second and
third prize. How many ways can the
3 people be chosen?
Question 13

16. You are ordering a triple-scoop
ice-cream cone. There are 18 flavors
to choose from and you don’t care
which flavor is on the top, middle, or
bottom. How many different ways can
you select a triple-scoop ice-cream
cone?
Question 14

17. An art gallery has 12 paintings
in storage. They have room to
display 4 of them, with each painting
in a different room. How many
possible ways can they display the 4
additional paintings.
Question 15

18. Permutation
GAGANI, MARYNEL C.

19. A permutation is an arrangement of things
in a certain order or the arrangement of
distinguishable objects without allowing
repetitions among the objects.
In general, if n is a positive integer, then n
factorial denoted by n! is the product of all
positive integers less than or equal to n.
n!=n•(n-1)•(n-2)•…•2•1
Permutation

20. Compute

21. The notation P(n,r) represents the number of
permutations (arrangements) of n objects taken r
at a time when r is less than or equal to n.
In general,
P(n,r) = n(n-1)(n-2)(n-3)…(n-r+1)
Permutation

22. Permutation Formula
)!(
!
),(
rn
n
rnPPrn



23. Permutation
This formula is used when a counting problem
involves both:
Choosing a subset of r elements from a set of n
elements; and
Arranging the chosen elements.

24. Permutation
EXAMPLE 1:
Suppose we wish to arrange n = 5 people {a, b,
c, d, e}, standing side by side, for a portrait. How
many such distinct portraits (“permutations”) are
possible?

25. Permutation
Solution:
There are 5 possible choices for which person stands
in the first position (either a, b, c, d, or e). For each of these
five possibilities, there are 4 possible choices left for who is in
the next position. For each of these four possibilities, there are
3 possible choices left for the next position, and so on.
Therefore, there are 5 × 4 × 3 × 2 × 1 = 120 distinct
permutations.
This number, 5 × 4 × 3 × 2 × 1 (or equivalently, 1 × 2 ×
3 × 4 × 5), is denoted by the symbol “5!” and read “5
factorial”, so we can write the answer succinctly as 5! =
120.

26. Permutation

27. Permutation
Examples: 6! = 6 × 5 × 4 × 3 × 2 × 1
= 6 × 5!
= 6 × 120 (by previous calculation)
= 720
3! = 3 × 2 × 1 = 6
2! = 2 × 1 = 2
1! = 1
0! = 1,

28. Permutation
EXAMPLE 2:
Now suppose we start with the same n = 5
people {a, b, c, d, e}, but we wish to make portraits of
only k = 3 of them at a time. How many such distinct
portraits are possible?

29. Permutation

30. Permutation

31. Answer Key
1. Combination
2. Permutation
3. Permutation
4. Permutation
5. Combination
6. Permutation
7. Permutation
8-12. Combination
13. Permutation
14. Combination
15. Permutation

32. Special Permutation when
letters must repeat
Example:
How many permutations of the word seem
can be made?
Since there are 4 letters, the total possible
ways is 4! IF each “e” is labeled differently. Also,
there are 2! Ways to permute e1e2. But, since
they are indistinguishable, these duplicates must
be eliminated by dividing by 2!.

33. Special Permutation when
letters must repeat

34. The number of permutations of n objects in
which k1 are alike, k2 are alike, etc.
!!...!!
!
321 p
rn
kkkk
n
P 
Special Permutation when
letters must repeat

35. Special Permutation when
letters must repeat
Find the permutations of the word Mississippi.
Number of Letters
– 11 – Total Letters
– 1 – M
– 4 – I
– 4 – S
– 2 - P
34650
)!2!4!4!1(
!11
