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.

Assalamulaikum ....
To all my student Selamat Hari Raya Maaf Zahir dan Batin
Hari Raya disambut setahun sekali
Sijil Pelajaran Malaysia seumur hidup sekali
Berikut adalah fail untuk semua pelajar cikgu membuat ulangkaji, jika ada masallah email cikgu di sahgi63@yahoo.com.my

Bahan intervensi maths
Bahan intervensi add maths
Modul back to basic
Modul bimbingan Add maths
Modul bimbingan Maths

3.Perisian membuat graph ( Zip Fail)

4. SUPER SODUKU KOKURU intermediate soduku

5 Career in maths (Fail PPT)

Bahan ini boleh membantu anda memahami konsep dalam Pembelajaran Additional Mathematics.Sila download dan menafaatkan.

You should highlight and understand the keywords given in the question because these keywords are the gist of the question. Some commonly found keywords are as follows

KEYWORD Calculate, Find, Determine
What is its implication?
The answer can be obtained after a few steps of determine calculation. Hence, the working must be shown clearly.

KEY WORD Write, state,
The answer can be worked out mentally. Hence, you can write down the answer without the working.

KEY WORD Hence
Usually you have to use the answer obtained form the previous section in your calculation.

KEY WORD: Sketch ,
Only a sketch of the graph is required. You can sketch the graph on a plain paper or a writing paper. Accurate plotting on the graph is not required.

KEY WORD : Prove,show,
The answer is usually given. You are required to show clearly the steps how you arrive at the answer. In this type of question, you must be familiar with the mathematical formulas, rules and laws.

KEY WORD : Express
No numerical answer is required. Answer are usually given in terms of variables.

1. Keep a list of things to remember - problems stressed by the instructor, definitions, terms, diagrams and graphs, formulas.
   Keep up with the work - some courses can be passed by cramming, but math isn’t one of them. Skills in math, as in sports, must be practiced.
   
2. Study copies of old exams, chapter tests from the book, or make up your own. Then practice them with the same limits as the real exam.
   Get a good night’s sleep before the test so that you are rested and alert; a quick review before the test should be a summary only.
   Arrive at the test early so that you can be relaxed when the exam begins.
   
3. Quickly look over the test and budget your time - don’t spend too much time on any single problem or section of the test.
   Do some work on each problem - try to work at least part of each problem because partial credit is better than none.
   
4. Check your answers and look for careless mistakes during the last few minutes of your test time (budget this important time).