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 nCr = n! r! (n – r)! 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 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: All Permutations All Combinations without repetitions 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: Find the number of different samples of 4 Find the number of different samples of 4 containing: (1) No Defectives (2) 1 Defective (3) 2 Defectives

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 ?