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Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com PDF Download

What's the Difference?

In English we use the word "combination" loosely, without thinking if the order of things is important. In other words:

  • "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

  • "The combination to the safe is 472". Now we do care about the order. "724" won't work, nor will "247". It has to be exactly 4-7-2.

So, in Mathematics we use more precise language:

  • When the order doesn't matter, it is a Combination.

  • When the order does matter it is a Permutation.

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

So, we should really call this a "Permutation Lock"!

In other words:

A Permutation is an ordered Combination.

 

Permutations

There are basically two types of permutation:

  1. Repetition is Allowed: such as the lock above. It could be "333".

  2. No Repetition: for example the first three people in a running race. You can't be first andsecond.

 

1. Permutations with Repetition

These are the easiest to calculate.

When a thing has n different types ... we have n choices each time!

For example: choosing 3 of those things, the permutations are:

n × n × n 
(n multiplied 3 times)

More generally: choosing r of something that has n different types, the permutations are:

n × n × ... (r times)

(In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.)

Which is easier to write down using an exponent of r:

n × n × ... (r times) = nr

Example: in the lock above, 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:

nr

where n is the number of things to choose from, 
and we choose r of them,
repetition is allowed, 
and order matters.

 

2. Permutations without Repetition

In this case, we have to reduce the number of available choices each time.

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com


Example: what order could 16 pool balls be in?

After choosing, say, number "14" we can't choose it again.

So, our first choice has 16 possibilites, and our next choice has 15 possibilities, then 14, 13, 12, 11, ... etc. And the total permutations are:

16 × 15 × 14 × 13 × ... = 20,922,789,888,000

But maybe we don't want to choose them all, just 3 of them, and that is then:

16 × 15 × 14 = 3,360

In other words, there are 3,360 different ways that 3 pool balls could be arranged out of 16 balls.

 

Without repetition our choices get reduced each time.

But how do we write that mathematically? Answer: we use the "factorial function"

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples:

  • 4! = 4 × 3 × 2 × 1 = 24

  • 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

  • 1! = 1

Note: it is generally agreed that 0! = 1. It may seem funny that multiplying no numbers together gets us 1, but it helps simplify a lot of equations.

So, when we want to select all of the billiard balls the permutations are:

16! = 20,922,789,888,000

But when we want to select just 3 we don't want to multiply after 14. How do we do that? There is a neat trick: we divide by 13!

16 × 15 × 14 × 13 × 12 ...13 × 12 ...  =  16 × 15 × 14

That was neat. The 13 × 12 × ... etc gets "cancelled out", leaving only 16 × 15 × 14.

The formula is written:

n!(n − r)!

where n is the number of things to choose from, 
and we choose r of them,
no repetitions,
order matters.

Example Our "order of 3 out of 16 pool balls example" is:

16!

 = 

16!

 = 

20,922,789,888,000

 = 3,360

(16-3)!

13!

6,227,020,800

(which is just the same as: 16 × 15 × 14 = 3,360)

Example: How many ways can first and second place be awarded to 10 people?

10!

 = 

10!

 = 

3,628,800

 = 90

(10-2)!

8!

40,320

(which is just the same as: 10 × 9 = 90)

Notation

Instead of writing the whole formula, people use different notations such as these:

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

Example: P(10,2) = 90


Combinations

There are also two types of combinations (remember the order does not matter now):

  1. Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)

  2. No Repetition: such as lottery numbers (2,14,15,27,30,33)

 

1. Combinations with Repetition

Actually, these are the hardest to explain, so we will come back to this later.

2. Combinations without Repetition

This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!

The easiest way to explain it is to:

  • assume that the order does matter (ie permutations),

  • then alter it so the order does not matter.

Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order.

We already know that 3 out of 16 gave us 3,360 permutations.

But many of those are the same to us now, because we don't care what order!

For example, let us say balls 1, 2 and 3 are chosen. These are the possibilites:

Order does matter

Order doesn't matter

1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1

1 2 3

So, the permutations have 6 times as many possibilites.

In fact there is an easy way to work out how many ways "1 2 3" could be placed in order, and we have already talked about it. The answer is:

3! = 3 × 2 × 1 = 6

(Another example: 4 things can be placed in 4! = 4 × 3 × 2 × 1 = 24 different ways, try it for yourself!)

So we adjust our permutations formula to reduce it by how many ways the objects could be in order (because we aren't interested in their order any more):

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

That formula is so important it is often just written in big parentheses like this:

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

where n is the number of things to choose from, 
and we choose r of them,
no repetition, 
order doesn't matter.

It is often called "n choose r" (such as "16 choose 3")

And is also known as the Binomial Coefficient.

Notation

As well as the "big parentheses", people also use these notations:

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

 

Just remember the formula:

n!r!(n − r)!

Example: Pool Balls (without order)

So, our pool ball example (now without order) is:

16!3!(16−3)! = 16!3! × 13!

= 20,922,789,888,0006 × 6,227,020,800

= 560

Or we could do it this way:

16×15×143×2×1 = 33606 = 560

 

It is interesting to also note how this formula is nice and symmetrical:

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

In other words choosing 3 balls out of 16, or choosing 13 balls out of 16 have the same number of combinations.

16!3!(16−3)! = 16!13!(16−13)! = 16!3! × 13! = 560

Pascal's Triangle

We can also use Pascal's Triangle to find the values. Go down to row "n" (the top row is 0), and then along "r" places and the value there is our answer. Here is an extract showing row 16:

1 14 91 364 ...
1 15 105 455 1365 ...
1 16 120 560 1820 4368 ...

 

1. Combinations with Repetition

OK, now we can tackle this one ...

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla.

We can have three scoops. How many variations will there be?

Let's use letters for the flavors: {b, c, l, s, v}. Example selections include

  • {c, c, c} (3 scoops of chocolate)

  • {b, l, v} (one each of banana, lemon and vanilla)

  • {b, v, v} (one of banana, two of vanilla)

(And just to be clear: There are n=5 things to choose from, and we choose r=3 of them.
Order does not matter, and we can repeat!)

Now, I can't describe directly to you how to calculate this, but I can show you a special techniquethat lets you work it out.

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

Think about the ice cream being in boxes, we could say "move past the first box, then take 3 scoops, then move along 3 more boxes to the end" and we will have 3 scoops of chocolate!

So it is like we are ordering a robot to get our ice cream, but it doesn't change anything, we still get what we want.

We can write this down as Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com (arrow means move, circle means scoop).

In fact the three examples above can be written like this:

{c, c, c} (3 scoops of chocolate):

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

{b, l, v} (one each of banana, lemon and vanilla):

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

{b, v, v} (one of banana, two of vanilla):

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

OK, so instead of worrying about different flavors, we have a simpler question: "how many different ways can we arrange arrows and circles?"

Notice that there are always 3 circles (3 scoops of ice cream) and 4 arrows (we need to move 4 times to go from the 1st to 5th container).

So (being general here) there are r + (n−1) positions, and we want to choose r of them to have circles.

This is like saying "we have r + (n−1) pool balls and want to choose r of them". In other words it is now like the pool balls question, but with slightly changed numbers. And we can write it like this:

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

where n is the number of things to choose from, 
and we choose r of them
repetition allowed, 
order doesn't matter.

Interestingly, we can look at the arrows instead of the circles, and say "we have r + (n−1)positions and want to choose (n−1) of them to have arrows", and the answer is the same:

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

So, what about our example, what is the answer?

Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com

There are 35 ways of having 3 scoops from five flavors of icecream

The document Permutation - Probability, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com is a part of the B Com Course Business Mathematics and Statistics.
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FAQs on Permutation - Probability, Business Mathematics and Statistics - Business Mathematics and Statistics - B Com

1. What is permutation in probability and statistics?
Ans. Permutation refers to the arrangement of a set of objects in a specific order. In the context of probability and statistics, it is used to calculate the number of possible arrangements of a given set of elements. The formula for permutation is nPr = n! / (n-r)!, where n represents the total number of objects and r represents the number of objects to be selected.
2. How is permutation different from combination in probability?
Ans. Permutation and combination are both mathematical concepts used in probability, but they differ in terms of the order of arrangement. Permutation considers the order of arrangement, while combination does not. In permutation, the arrangement ABC is considered different from BAC, while in combination, the order does not matter, so ABC and BAC are considered the same combination.
3. How can permutation be used in business mathematics?
Ans. Permutation is commonly used in business mathematics to calculate the number of ways objects can be arranged or ordered, which can have practical applications. For example, in inventory management, permutation can be used to determine the number of ways products can be arranged on a shelf, helping businesses optimize their display strategies. It can also be used in scheduling and resource allocation, where the order of tasks or resources can impact efficiency and productivity.
4. What is the significance of permutation in statistics?
Ans. Permutation plays a significant role in statistics, particularly in hypothesis testing and experimental design. In hypothesis testing, permutation tests are used to generate the null distribution when the assumptions of parametric tests are violated. This allows for more robust statistical analysis, especially when dealing with non-normal or non-independent data. Additionally, permutation tests are also used in experimental design to randomize the assignment of treatments to subjects, ensuring unbiased results.
5. Can permutation be used to calculate the probability of an event?
Ans. Yes, permutation can be used to calculate the probability of an event when all outcomes are equally likely. By determining the total number of possible outcomes (n) and the number of favorable outcomes (r), the probability can be calculated using the formula P = r / n. However, it is important to note that this method assumes each outcome is equally likely, which may not always be the case in real-world scenarios.
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