A-level revision

Factorials, Combinations and Permutations

One of the hardest parts of Statistics 1 can be an understanding of the distinction between the different types of ways of ordering and choosing objects

This page offers concise examples of the main types you will be expected to know and be able to use

Factorials

Suppose we want the total number of ways of arranging  different items.

 

1) How many ways can we arrange  books on a bookshelf

Answer:

 

2) If 15 people run a race, how many different ways of arranging 1st to 15th are there?

Answer:

 

Permutations

1) Suppose we have people running a race and we care about the ways of getting  

There are obviously quite a lot of options. If we call the people A B C D E F G H I J three possible options out of many more include:

 

1st

2nd

3rd

4th

5th

A

B

C

D

E

B

A

C

D

E

F

G

H

I

J

 

The first two options are different BECAUSE a different person comes first and second

The third option is different BECAUSE it used an entirely different set of people

 

 

2) We have every letter of the alphabet on a tile. We pick out 4 at random and once a tile has been picked out it cannot be used again. How many different four letter random words can we write down?

 

This is comparable to the problem above because writing down ACBD is different to ABCD

Therefore the answer is

 

Another way to think about is, we have 26 options for the first tile, 25 for the second, 24 for the third and 23 for the fourth tile so the number of words equals

 

 

 

 

 

 

Combinations (arguably the most important one)

 

Choosing from a list of different items

 

1) Suppose I have 10 people in a room, again we will call them A B C D E F G H I J

If we want the number of ways of choosing 5 people out of this group there are a lot less options than before. This is because choosing people ‘ABCDE’ is NOW THE SAME as ‘BACDE’.

 

We do not care about the order of the people we pick – we only care about the end result of WHO WE HAVE PICKED.

 

The total number of ways of choosing five people out of ten is

 

Arranging TWO sets of identical items - BINOMIAL

 

1) Suppose I toss a coin ten times. How many ways of getting 4 heads and 6 tails are there?

 

Some options include:

 

 

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

Option 1

H

H

H

H

T

T

T

T

T

T

Option 2

H

H

 H

T

H

T

T

T

T

T

 

And so on…there are way too many to even begin to list them

 

However, we have a total of 10 results. 4 are identical HEADS and 6 are identical TAILS.

 

The total number of ways can be therefore found by working out either  or

 

2) There are 365 days in a year. How many different ways are there for it to be dry on 100 days of the year (which means not dry on 265 days of the year)

 

The answer is either  or

 

 

Multiple combinations

I want to make a band of two drummers, three guitarists, 2 singers and a director.

 

I have a pool of 10 drummers, 6 guitarists, 11 singers and 1 director from which to choose from

 

We work out the number of ways of choosing each particular sub-group and multiply all the answers together.

 

The total number of ways off arranging the band is therefore:

 

Drummer

 

Guitarist

 

Singer

 

Director

 

2 from 10

 

3 from 6

 

2 from 11

 

1 from 1

 

 

 

 

 

 

 

 

 

Applying to situations in probability

1) 30 books are placed onto a shelf. What is the probability they are placed in alphabetical order?

 

Solution: There are a total of  ways of arranging them. Only one is in alphabetical order, so:

 

 

2) 15 horses run a race. What is the probability of getting 1st, 2nd and 3rd place in the correct order?

 

Solution 1: There are  different ways of picking 1st, 2nd and 3rd, so:

 

 

Solution 2: Multiply the probability of guessing 1st place, then 2nd place, then 3rd place

 

3) Suppose we have the letters J A M E S on a set of tiles and we pick out 3 tiles:

 

a)

 

b)

 

4) If we have a spinner with two colours such that

 

If we spin the spinner 20 times.