Menu

Traditional Menu Other Menu Nasi Lemak Beef Burger

Roti Canai Potatoes Chip Capati French Bread Mee Goreng

Drink : Coffee, Tea, Coke and Pepsi

Questions? How many different breakfast consists of

traditional menu and drink. How many different breakfast consists of

other menu and drink. How many different breakfast consists of

traditional, other menu and drink. How many different breakfast consists of

traditional menu and an optional drink.

Basic principlesMultiplication principle

If an activity can be performed in k successive steps, Step 1 can be done in n1 ways

Step 2 can be done in n2 ways … Step k can be done in nk ways

Then: the number of different ways that the

activity can be performed is the product

n1n2…nk

More Questionsa. How many strings of length 4 can be formed using the letter ABCDE if

repetitions are not allowed

b. How many strings of part (a) begin with the letter B?

c. How many strings of part (a) do not begin

with the letter B?

Virus Attack After the virus sent the email to the first

50 addresses each of those recipients then sent email to 50 addresses. This means 50 *50 =2500 additional recipients. Each of the recipients then sent email to 50 addresses. This means 50*50*50=125000 additional recipients. Thus after three iterations 125000+2500+50+1=127551 copies of the message had been sent.

Examples How many eight-bit strings begin either

101 and 111?

How many eight-bit strings begin 1100?

How many eight-bit strings begin and end with 1?

Addition principleLet X1, X2,…, Xk be a collection of k pair-

wise disjoint sets, each of which has nj elements, 1 < j < k, then the union of those sets

k

X = Xj

j =1

has n1 + n2 + … + nk elements

Example A six-person committee composed of A, B,

C, D, E and F is to select a chairperson, secretary and treasurer. How many ways can this be done? How many ways can this be done if either A or

B must be chairperson? How many ways can this be done if E must

hold one of the offices? How many ways can this be done if both D and

F must hold office?

Example 2 How many ways can we select two books

from different subjects among five distinct computer science books, three distinct mathematics books and two distinct art books?

Permutations and combinations

A permutation of n distinct elements x1, x2,…, xn is an ordering of the n elements. There are n! permutations of n elements.

Example: there are 3! = 6 permutations of three elements a, b, c:

abc bac cab

acb bca cba

Examples How many permutations of the letter

ABCDEF contain the substring DEF?

How many permutations of the letter ABCDEF contain the letter DEF in any order?

r-permutationsAn r-permutation of n distinct elements is an

ordering of an r-element subset of the n elements x1, x2,…, xn

For r < n the number of r-permutations of a set with n distinct objects is

P(n,r) = n(n-1)(n-2)…(n-r+1)

Examples The number of 2-permutations of X={a, b,

c} is P(3,2)=3!/(3-2)!=3!/1!=3*2*1/1=6

ab, ac, ba, bc, ca, cb

How many 3-permutations are there of a, b, c, d? P(5,3)=5!/(5-2)!=5!/3!=20

How many 5-permutations are there of 11 distinct object2? P(11,5)=11!/(11-5)!

More Examples How many different ways are there to

select 4 different players from 10 players on a team to play four tennis matches where the matches are ordered? P(10,4)=10!/(10-4)!=5040

Let S={1,2,3,4,5} List all the 3-permutations of S?

P(5,3)=5!/(5-3)!=5.4.3=60

Combinations

Let X = {x1, x2,…, xn} be a set containing n

distinct elements An r-combination of X is an unordered

selection of r elements of X, for r < n The number of r-combinations of X is the

C(n,r) = n! / r!(n-r)!

Examples Let S={1,2,3,4,5}

List all the 3-combinations of S C(5,3)=5!/(3!)(5-3)!

How many ways are there to select 5 players from a 10-members tennis team to make a trip to a match? C(10,5)

How many ways can we select a committee of two women and three men from a group of five women and six men? First task : C(5,2) Second Task : C(6,3)

Exercise A club consisting of six men and seven

women.

How many ways can we select a committee of five persons?

How many ways can we select a committee of three men and four women?

The pigeonhole principle First form: If k < n and n pigeons fly into k

pigeonholes, some pigeonhole contains at least two pigeons.

Example Show that if any five numbers from 1 to 8

are chosen, then two of them will add to 9. Construct four different sets, A={1,8}

B={2,7}, C={3,6}, D={4,5}. Each of the five numbers chosen must belong to one of these sets. Since there are only four sets the pigeonhole principle tells us that two of the chosen number belong to the same set and these numbers add up to 9

Examples If eight people are chosen in any way from some

group at least two of them will have been born on the same day of the week.

Shirt numbered consecutively from 1 to 20 are worn by the 20 members of a bowling league. When any three of these numbers are chosen to be a team the league proposes to use the sum of their numbers as a code number for the team. Show that if any eight of the 20 are selected then these eight one may form at least two different teams having the same code number.

Examples

Show that if 30 dictionaries in a library contains a total of 61327 pages, then one of the dictionaries must have at least 2045.

Show that if any 30 people are selected then one may choose a subset of five so that all five were born on the same day of the week

Second form of the pigeonhole principle

If X and Y are finite sets with |X| > |Y| and f : X Y is a function, then f(x1) = f(x2) for some x1, x2 X, x1 x2.

Third form of the pigeonhole principle

If X and Y are finite sets with |X| = n, |Y| = m and k = n/m, then there are at least k values a1, a2,…, ak X such that f(a1) = f(a2) = … f(ak).

Example: n = 5, m = 3k = n/m = 5/3 = 2.