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.

EXAMPLEThe 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 :

Without Replacement : n + r – 1 Cr = (n + r – 1)! r! (n – 1)!

nCr = n! r! (n – r)!

Problem 1A 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 2List following of vowel letters taken 2 at

a time:a. All Permutationsb. All Combinations without repetitionsc. 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 72 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) No Defectives (2) 1 Defective (3) 2 Defectives

Problem 9A 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 13If 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 15How many different license plates

showing 5 symbols, namely 2 letters followed by 3 digits, could be made ?