To understand this concept for the competitive exams we give you brief explanations to write exames, they mostly ask in APPSC, DSC, TET, CONSTABLE SI and RAILWAYS. Now What is the difference between Combinations and Permutations. when the other doesn’t matter it is a combination, then order matters in a permutation. A permutation is the ordered combination. So we should have really called it a “permutation lock”. In other words, we called permutations the order of combinations. To help you to remember, think the permutation is a position.
There are basically two types of permutation:
- Repetition is Allowed: such as the lock at one. It could be “333”.
- No Repetition: for example the first three of them people are running a race. You can’t be first and second.
Permutations with Repetition
These are the very easiest to calculate. When the things have n different types then we dont have n choices each time!
For example: choosing 3 of those things, the permutations are: n × n × n then the n multiplied 3 times.
More generally: choosing r of something that has n different types, the permutations are: n × n × … in r times
Whereas in other words, there are n possibilities for the first choice, Then there are n possibilities for the second choice, and so on, multiplying each time.
It is easier to write down using an exponent of r: n × n × … r times = nr
Look at the Above example: There are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 3 of them:10 × 10 × … (3 times) = 103 = 1,000 permutations
So, the formula is simply: n2 where n is the number of things to choose from.
Permutations without Repetition
In this situation, we have to reduce the number of available selections from it.
Example: order could 16 pool balls be in?
After choosing a number 14 we can’t choose it again. So, then our first choice to select the 16 possibilities, and then our next choice to select the 15 possibilities, then 14, 13, 12, 11, … etc.
These total permutations are:16 × 15 × 14 × 13 × . = 20,922,789,888,000 But we don’t want to choose them all, just 3 of them, and that is :16 × 15 × 14 = 3,360 In other words, there are 3,360 different ways that the 3 pool balls could be arranged out of 16 balls.
There are also two types of combinations (remember the order doesn’t matter now):
- Repetition is Allowed: such as coins in your pocket are 5,5,5,10,10
- No Repetition: such as lottery numbers are 2,14,15,27,30,33
Combinations with Repetition
Actually, these are the hardest to explain the combinations with repetition, so we will come back to this later.
Combinations without Repetition
This is how lotteries work. Numbers are drawn from it one at a time and if we have the lucky numbers no matter what the order we win!
The easiest way to explain it is: Example for the pool ball let as know how and which pool balls are chosen, not the order,
We know that 3 out of 16 gave us and 3,360 permutations. But now many of those are the same to us, because we don’t care what the order is.
For example, let us say balls 1, 2 & 3 are selected.
These are the possibilities:
Oder does matter
1 2 3
1 3 2
2 1 3 order doesn’t matter 123
2 3 1
3 1 2
3 2 1
So, the permutations have 6 times as many as. In fact there is an easy way to work out how many ways “1 2 3” could be placed in order, and we have talked about it already. The answer is: 3! = 3 × 2 × 1 = 6
One more example: 4 things can be placed in the 4! = 4 × 3 × 2 × 1 = 24 in different ways, try this for yourself!
So we have to adjust our permutations formula to reduce it by how many ways the objects could be in order.
But knowing how these formulas work is only half the struggle. And knowing how the Similar the real-world situation is. But at least now we know how to calculate all 4 variations in the combinations and permutations.