Combinations

Combinations

Combinations

What is the difference between permutations and combinations?

• A combination is the number of possible arrangements of a set of objects when the order of the arrangements does not matter

  • On the other hand a permutation is when the order of arrangement does matter

• A combination will be finding the number of ways to choose r out of n items

  • The order in which the r items are chosen is not important

  • For example if we are choosing two letters from the word CAB, AB and BA would be considered the same combination but different permutations

How do we find r combinations of n items?

• If we want to find the number of ways to choose 2 out of 3 different objects, but we don’t mind the order in which they are chosen, then we could find the number of permutations of 2 items from 3 and then divide by the number of ways of arranging each combination

  • For example if we want to choose 2 letters from A, B and C

    • There are 6 permutations of 2 letters:

    • AB, BA, AC, CA, BC, CB

    • For each combination of 2 letters there are 2 (2 × 1) ways of arranging them

    • (for example, AB and BA)

    • So divide the total number of permutations (6) by the number of ways of arranging each combination (2) to get 3 combinations

• If we want to find the number of ways to choose 3 out of 5 different objects, but we don’t mind the order in which they are chosen, then we could find the number of permutations of 3 items from 5 and then divide by the number of ways of arranging each combination

  • For example if we want to choose 3 letters from A, B, C, D and E

    • There are 60 permutations of 3 letters:

    • ABC, ACB, BAC, BCA, CAB, CBA, ABD, ADB, etc

    • For each combination of 3 letters there are 6 (3 × 2 ×1) ways of arranging them (for example, ABC, ACB, BAC, BCA, CAB and CBA)

    • So divide the total number of permutations (60) by the number of ways of arranging each combination (3! = 6) to get 10 combinations

• If we want to find the number of ways to choose r items out of n different objects, but we don’t mind the order in which they are chosen, then we could find the number of permutations of r items from n and then divide by the number of ways of arranging each combination

• Recall that the formula for r permutations of n items is 

  • ${}_{ }{}^n{\mathrm{P}}_r= \frac{n!}{(n - r)!}$

• This would include r! ways of repeating each combination

• The formula for r combinations of n items is

  • $\frac{{}_{ }{}^n{\mathrm{P}}_r}{r!}= \frac{n!}{\left(n - r\right)! r!}$

• The function $\frac{n!}{\left(n – r\right)! r!}$ can be written as ${}_{ }{}^n{\mathrm{C}}_r$or $\left(\begin{array}{c}n\\ r\end{array}\right)$ and is often read as ‘n choose r’

  • Make sure you can find and use this button on your calculator

• The formulae for permutations and combinations satisfy the following relationship:

• ${}_{ }{}^n{\mathrm{C}}_r=\frac{{}_{ }{}^n{\mathrm{P}}_r}{r!}$

What do I need to know about combinations?

• The formula ${}^{n}C_r=\frac{n!}{(n-r)!r!}$ is also known as a binomial coefficient

• ${}^{n}C_n={}^{n}C_0=1$

  • It is easy to see that there is only one way of arranging n objects out of n and also there can only be one way of arranging 0 objects out of n

  • By considering the formula for this, it reinforces the fact that 0! Must equal 1

• The binomial coefficients are symmetrical, so ${}^{n}C_r={}^{n}C_{n-r}$

  • This can be seen by considering the formula for ${}^{n}C_r$

  • ${}^{n}C_{n-r}=\frac{n!}{\left(n-r\right)!(n-(n-r\left)\right)!}=\frac{n!}{r!(n-r)!}={}^{n}C_r$

How do I know when to multiply or add?

• Many questions will ask you to find combinations of a group of different items from a bigger group of a specified number of those different items

  • For example, find the number of ways five questions could be chosen from a bank of twenty different pure and ten different statistics questions

  • The hint in this example is the word 'chosen', this tells you that the order in which the questions are chosen doesn't matter

• Sometimes questions will have restrictions,

  • For example there should be three pure and two statistics chosen from the bank of questions, 

  • Or there must be at least two pure questions within the group

• If unsure about whether to add or multiply your options, ask yourself if A and B are both needed, or if A or B is needed

  • Always multiply if the answer is and, and add if the answer is or

  • For example if we needed exactly three pure and two statistics questions we would find the amount of each and multiply them

  • If we could have either five statistics or five pure questions we would find them separately and add the answers

• Probabilities can be found with combinations questions by finding the number of options a selection can be made in a particular way and dividing that by the total number of options