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Counting Methods and their Uses
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30 Apr 2015, 08:00
We are starting this thread to shed more light on the various counting methods and their uses in different situations.
1. Permutation v/s Combination: Which one to use?
The basic difference between a combination and a permutation is that while the former is just a way of selecting something, the latter is a way of selecting as well as arranging it.
Q1. In how many ways can you select 2 letters out of A, B, C and D?
Here, we are only interested in choosing 2 letters. Each set of 2 different letters can be considered a combination.
AB, BC, CD, AD, AC and BD are the 6 different combinations possible.
But if the question were posed slightly differently, as –
Q2. In how many ways can 2 letters out of A, B, C and D be placed in a line?
Now we are interested in choosing as well as arranging 2 letters because AB, placed in a line, would look different from BA.
So the different permutations possible are the first 6 we saw earlier, namely AB, BC, CD, AD, AC and BD, as well as the arrangements of each set, namely BA, CB, DC, DA, CA and DB. So total ways = 6 * 2 (number of ways 2 letters can be arranged)
This is what the difference between nCr and nPr is, too.
While nPr denotes the ways of selecting and arranging r objects out of n, nCr is just concerned with their selection. Since for each selection we get r! arrangements (the number of ways of arranging r objects is r!), the relationship between the two is \(nCr = \frac{nPr}{r!}\)
So if you ever get stuck on a question and can’t decide whether you are looking for a permutation or a combination, just decide whether the final arrangement of your selection makes a difference to that situation or not.
Let’s take a few examples to understand this point.
Q3. In how many ways can you select a tennis doubles pair out of 6 players?
Ans: The arrangement isn’t important. AB is same as BA. Go with 6C2.
Q4. In how many ways can 5 people sit in 5 chairs?
Ans: The arrangements matter. ABCDE is different from BCDEA. Go with 5P5 (5!).
Q5. In how many ways can a 3striped flag be made out of 7 colors?
Ans: Here we need to choose as well as arrange 3 colors out of 7. Red, green, blue would look different from blue, green and red. Go with 7P3.
Takeaway: Use permutations if arrangements are important. Use combinations otherwise.
Connecting nCr, nPr and basic counting
In the end, permutations, combinations or just basic counting are all different ways of reaching the right answer. Any method can be used in any situation as long as we understand what it means.
To demonstrate that, let’s take a look at Q5 again.
Q5. In how many ways can a 3striped flag be made out of 7 colors?
Ans: This can be done in 7P3 ways as seen earlier.
Or choose the 3 colors using 7C3. Then arrange them in 3! ways. 7C3 * 3! is same as 7P3.
Or you can simply say: Ways in which we can choose the first color = 7 Ways in which we can choose the second color = 6 Ways in which we can choose the third color = 5
Total number of ways: 7*6*5 = 7P3 = 7C3 * 3!
Go ahead and try this on other questions of counting that you come across. Solving the same question in different ways will boost your confidence and make your grasp on these concepts much stronger.
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