Permutations and Combinations (Add Maths)

Working on “permutation” and “combination” can be confusing isn't it?  When presented with a problem, firstly you will need to decide which one's which, otherwise your misunderstanding of the concept will give you the wrong solution.

Here’s an easy way to remember: which sounds more complex than the other?  "permutation sounds complicated"! And it is more complicated than combination.

In permutations, all details need to be considered. As a example, arranging ABCD and DBCA is not the same thing in permutations. Or seating Marian, Janet, Selvam and John in that order in a row is different from John, Janet, Selvam and Marian arranged in a row.

On the other hand, with Combinations, the details don’t matter. Whether Marian is seated next to John or Selvam or Janet in a row, is the same as having Janet, Marian, John and Selvam seated to each other in a row.

Now, do you get the meaning?

Combination is just like having a rojak. You need to just put in the ingredients without having to think which one comes first or last.

Rojak --pergghh.. tiba-tiba terasa petang ni kena cari rojak dan cendol..

Let's try solve a problem:

Six people – Hasif, Imran, Haziq, Aiman, Daniel and Amirul   want to ride in a boat that can hold only 4 passengers. 

How many different groups of 4 passengers can ride in the boat?

Step 1: Write out a single order.

Hasif, Imran, Haziq, Aiman

Step 2: Now rearrange the order. Did changing the order of the items change the outcome? If so, then order matters.

Haziq, Imran, Aiman, Hasif -<------now we have different order but the same 4 passengers.

Order does NOT matter for this problem. Use combinations.

Therefore, the answer is 6C4 = 15

Let's try another one 

In a group of 8 boys and 5 girls, five children are to be selected. In how many different ways can they be selected such that at least one boy should be there? 

We may have (1 boy and 4 girls) or (2 boys and 3 girls) or (3 boys and 2 girl) or (4 boys and 1 girls) or (5 boys)

Required number of ways:

(8C1 X 5C4 ) + (8C2 x 5C3) + (8C3 x 5C2) + (8C4 x 5C1)  + (8C5) 

= (8 x 5) + (28 x 10) + (56 x 10) + (70 x 5) + 56

= 40 + 280 + 560 + 350 + 56

= 1,286

In how many different ways can the letters of the word 'BERANI' be arranged in such a way that the vowels occupy only the odd positions?

There are 6 letters in the word given above, and there are 3 vowels and 3 consonants in it.
Let us mark these positions as follows

(1), (2), (3), (4), (5) , (6)

3 vowels can be placed at any of the three places out 6, marked 1, 3, 5.
Number of ways of arranging the vowels = ³P₃ = 3! = 6.
3 consonants can be arranged at the remaining 3 positions.
Number of ways of these arrangements = ³P₃ = 3! = 6.
Total number of ways = (6 x 6) = 36

.

Image credit: rojak