COMBINATIONS
AND
PERMUTATIONS
REVISION PROBABILITY
A coin is biased so that a head is twice as
likely to occur as a tail. If the coin is tossed
three times, what is the probability of
getting two tails and one head?
Three Sigma Rule
Three-sigma rule, or empirical rule states
that for a normal distribution, nearly all values
lie within 3 standard deviations of the mean.
EXAMPLE
The scores for all students taking SAT (Scholastic
Aptitude Test) in 2012 had a mean of 490 and a
Standard Deviation of 100:
• What percentage of students scored between 390
and 590 on this SAT test ?
• One student scored 795 on this test. How did this
student do compared to the rest of the scores?
• NUST only admits students who are among the
highest 16% of the students in this test. What score
would a student need to qualify for admission to
the NUST?
Permutation
• A permutation is an arrangement of all or part
of a set of objects.
• Number of permutations of n objects is n!
• Number of permutations of n distinct objects
taken r at a time is
nPr = n!
(n – r)!
• Number of permutations of n objects arranged
is a circle is (n-1)!
Permutations
• The number of distinct permutations of n
things of which n1 are of one kind, n2 of a
second kind, …, nk of kth kind is
n!
n1! n2! n3! … nk!
Combinations
• The number of combinations of n distinct objects
taken r at a time is
With Replacement :
nCr
=
n!
r! (n – r)!
Without Replacement : n + r – 1 Cr = (n + r – 1)!
r! (n – 1)!
Problem 1
A showroom has 12 cars. The showroom
owner wishes to select 5 of these to
display at a Car Show. How many different
ways can a group of 5 be selected ?
Problem 2
List following of vowel letters taken 2 at
a time:
a. All Permutations
b. All Combinations without repetitions
c. All Combinations with repetitions
Problem 3
In how many ways can we assign 8
workers to 8 jobs (one worker to each
job and conversely) ?
Problem 7
2 items are defective out of a lot of 10
items:
a. Find the number of different
samples of 4
b. Find the number of different samples
of 4 containing:
(1)
(3)
No Defectives
2 Defectives
(2)
1 Defective
Problem 9
A box contains 2 blue, 3 green, 4 red
balls. We draw 1 ball at random and put
it aside. Then, we draw next ball and so
on. Find the probability of drawing, at
first, the 2 blue balls, then 3 green ones
and finally the red ones ?
Problem 11
Determine the number of different
bridge hands (A Bridge Hand consists of
13 Cards selected from a full deck of 52
cards)
Problem 13
If 3 suspects who committed a
burglary and 6 innocent persons are
lined up. What is the probability that a
witness who is not sure and has to pick
three persons will pick 3 suspects by
chance? That person picks 3 innocent
persons by chance?
Problem 15
How many different license plates
showing 5 symbols, namely 2 letters
followed by 3 digits, could be made ?