Permutations and Combinations

Permutations
When the order does matter it is called Permutation. In other words, a Permutation is an ordered Combination.

We should really call this a "Permutation Lock"!

There are basically two types of permutation:

• Repetition is Allowed: such as the lock above. It could be "00000".
• No Repetition: for example the first three people in a running race. You can't be first and second.

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)

So, the formula is:
n^r (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.

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 possibilities, 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.

So, the formula is:

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


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

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

• Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)
• No Repetition: such as lottery numbers (2,14,15,27,30,33)

1. Combinations with Repetition
Let us say there are five flavours of ice cream: banana, chocolate, lemon, strawberry and vanilla.

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

Let's use letters for the flavours: {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)

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

There are 35 ways of having 3 scoops from five flavours of ice cream.

2. Combinations without Repetition
 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 possibilities:

So, the permutations have 6 times as many possibilities.
So we adjust our permutations formula to reduce it by how many ways the objects could be in order. That formula is so important it is often just written in big parentheses like this:

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

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Mean vs Variance vs Standard Deviation

Mean: The mean is the average of the numbers.
Variance: Variance is nothing but the average of the squares of the deviations.
Standard Deviation: Standard Deviation is the square root of the numerical value obtained while calculating variance.

Definition of Variance
In statistics, variance is defined as the measure of variability that represents how far members of a group are spread out. It finds out the average degree to which each observation varies from the mean. When the variance of a data set is small, it shows the closeness of the data points to the mean whereas a greater value of variance represents that the observations are very dispersed around the arithmetic mean and from each other.

Definition of Standard Deviation
Standard deviation is a measure that quantifies the amount of dispersion of the observations in a dataset. The low standard deviation is an indicator of the closeness of the scores to the arithmetic mean and a high standard deviation represents; the scores are dispersed over a higher range of values.

Key Differences Between Variance and Standard Deviation
The difference between standard deviation and variance can be drawn clearly on the following grounds:

1. Variance is a numerical value that describes the variability of observations from its arithmetic mean. Standard deviation is a measure of dispersion of observations within a data set.
2. Variance is nothing but an average of squared deviations. On the other hand, the standard deviation is the root mean square deviation.
3. Variance is denoted by sigma-squared (σ2) whereas standard deviation is labelled as sigma (σ).
4. Variance is expressed in square units which are usually larger than the values in the given dataset. As opposed to standard deviation which is expressed in the same units as the values in the set of data.
5. Variance measures how far individuals in a group are spread out. Conversely, Standard Deviation measures how much observations of a data set differs from its mean.

Illustration

Mean

Variance

Standard Deviation

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