MOZAC MATHS DEPARTMENT

Permutations and combinations

When we talk of permutations and combinations in everyday talk we often use the two terms interchangeably. In mathematics, however, the two each have very specific meanings, and this distinction often causes problems.
In brief, the permutation of a number of objects is the number of different ways they can be ordered; i.e. which is first, second, third, etc. If you wish to choose some objects from a larger number of objects, the way you position the chosen objects is also important. With combinations, on the other hand, one does not consider the order in which objects were chosen or placed, just which objects were chosen. We could summarise permutations and combinations (very simplistically) as

Permutations - position important (although choice may also be important)

Combinations - chosen important,
which may help you to remember which is which.

THE IMPORTANT DIFFERENCE
As mentioned above, there is an important difference between permutations and combinations. In this case, for permutations the order of events is important: order 1 is different from order 2. For combinations, however, it does not matter which picture was hung first. In this example there are two permutations (A, B ≠ B, A), but only one combination (A, B = B, A).
Another way that you may find useful to help you remember is to consider a combination lock. On combination locks you have to turn dials with numbers on so a particular number is given, e.g. '1, 2, 3, 4'. But they do not unlock when if the order is changed (e.g. 2, 1, 3, 4). In this case the order is important. So combination locks should not be called combination locks but 'permutation' locks.

In the same way that permutations have shorthand, combinations have similar shorthand. All we have to do is divide the number of permutations by the number of permutation in each set. So, the right-hand side of the following equation is the same as the equation for the number of permutations except for an additional r! term in the divisor (which corrects for the number of permutations of each set). Note, also, that the P (for permutation) is replaced by C (for combination).

If you have a scientific calculator you should see these labelled (on some calculators they are separate keys, on others they are second-function keys). In general, you enter the number of items to choose from (n) then the nCr or nPr button and then the number of items to choose (r).

Permutation - arrangement - example
Question 1 :   A family of 4 brothers and 3 sisters is to be arranged in a row for a photograph. In how many ways can they be seated if all the sisters are together?
Answer:    Let B₁,B₂,B₃,B₄ denote the brothers and S₁,S₂,S₃ denote the sisters. Since the sisters are to be seated together for a photograph, consider all the sisters as one unit or entity. Then B₁,B₂,B₃,B₄,S can be arranged to sit in 5! ways. The sisters can be arranged among themselves in 3! ways. Since the two events are independent, the total number of arrangements = 5!.3! = 720 ways.

Question 2:   In how many ways can a consonant and a vowel be chosen out of the letters in the word COURAGE?
Answer:    There are three consonants (C, R, G) and four vowels (A, E, O, U) in the word COURAGE.
With the consonants, we may choose any one of the 4 vowels. It can be done in 4 ways. There are three consonants.
\The total number of ways will be 4 x 3 = 12.

Question 3:   How many arrangements can be made out of the letters of the word DRAUGHT, the vowels never being separated?
Answer:    There are 7 letters in the word DRAUGHT, the two vowels are A and U. Since, the vowels are not to be separated, AU can be considered as one entity. Therefore, the number of letters will be 6 instead of 7. The permutations will be P(6,6) = 6! ways.
But the two vowels A and U can be arranged in two ways, i.e. AU and UA.
\The required number of arrangements = 2!.6! = 1440 ways.

Question 4:( Out of SPM Syllibus)   Find the number of arrangements that can be made out of the letters i) ASSASSINATION ii) GANESHPURI.
Answer:    i) The word ASSASSINATION consists of
A's = 3, S's = 4, I's = 2,
N's = 2, T's = 1, O's = 1
The total number of letters is 13 letters.


ii) The word GANESHPURI consists of 10 distinct letters.
The number of permutations is 10!.
Question 5 : ( out of SPM Syllibus)   How many different arrangements can be made out of the letters in the expression a³b²c⁴, when written at full length?
Answer:    There are 3 + 2 + 4 = 9 letters.

Standard form
Quadratic expressions
Sets
Mathematical reasoning
Straight lines
Statistics
Probability
Circle
Elevations
Plane 3D

SOLID GEOMETRY
CIRCLE, AREA AND PERIMETER
QUADRATIC EXPRESSIONS AND EQUATIONS.

SOLID GEOMETRY
CIRCLE, AREA AND PERIMETER
QUADRATIC EXPRESSIONS AND EQUATIONS.
SETS
INEQUALITIES
MATHEMATICAL REASONING
THE STRAIGHT LINE
STATISTICS
LINES AND PLANES IN 3-DIMENSIONS
GRAPHS OF FUNCTIONS
TRANSFORMATIONS
MATRICES
GRADIENT AND AREA UNDER A GRAPH
PROBABILITY
PLAN AND ELEVATION
EARTH AS A SPHERE

10. Praktis 10 dengan Skema 10