Combination: A collection of things in which order does not matter.
Notation: ncr
Formula: n!/r!*(n-r)!
Permutation: All possible arrangements of a collection of things where the order is important.
Notation: npr
Formula: n!/(n-r)!
The difference between Permutation and Combination is arrangements. A combination is generally selection without arrangement whereas Permutation is the selection with an arrangement.
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WHAT IS ARRANGEMENT?
Arrangement is number of possible orders.
Consider the question.
In how many ways the Letter "RAM" can be arranged.
So total there are 6 ways or 3! ways. Therefore N distinct things can be arranged in N! ways, but what if we have some identical and some distinct things.
Consider an example: In how many the word "RAMA" can be arranged.
when we have some identical things then arrangement becomes N!/m! where m is the number of things which are identical. Here we have 2 identical things which are 2 As.
We have looked at what arrangement is, Now lets come back to Permutation and Combination.
Question: In how many ways we can select 2 balls out of 5 balls of distinct colors ( red, blue, green, yellow, black).
This question only asks for selection.
So number of ways = 5c2
= 10 ways
but what if the question asks for selection and arrangement.
Then the number of ways would have been 5p2, which is 20
Let's have a different angle to this question if we are able to select 2 balls and use our knowledge of arrangements then we would be able to satisfy both the condition which is selection and arrangement.
We know we could select the ball in 10 ways, out of this 10 ways one selection would be red and blue, in how many ways we can arrange the ball red and blue is 2! so every combination in the 10 could be arranged in 2! ways. therefore selection and arrangement would be 10! * 2!. In this question 2 is r and 10 is ncr . therefore npr becomes ncr * r!.
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