7.15 Mixed combinations and permutations

Identifying whether a problem is asking for a combination or permutation, comes down to this question: Is the order important?

If the answer is yes and order matters, then it is a permutation (like the lock on a safe). 

If it is a permutation then we can use the Fundamental Counting Principle (boxes) or the notation and formula below.

$P(n,r)=$P(n,r)=$\nPr{n}{r}$nPr$=\frac{n!}{(n-r)!}$=n!(n−r)!​

 

If the answer is no and order does not matter - then it is a combination (like members of committee with no particular positions).

If a combination then we can use the notation and formula

$C(n,r)=$C(n,r)=$\nCr{n}{r}$nCr$=\frac{n!}{r!(n-r)!}$=n!r!(n−r)!​

Worked example

Question 1

Would the following be examples of permutations or combinations?

A) The number of zip codes which start with a 6 and end with an odd number.

Think: Does the order matter?

Do: 60001 and 60100 are two different zip codes, so order does matter. This would be a permutation.

B) The number of ways $12$12 people can be put into two groups of $6$6.

Think: Does the order matter?

Do: A group with people $1,2,3,4,5$1,2,3,4,5, and $6$6 is the same as a group with people $6,5,4,3,2$6,5,4,3,2, and $1$1, so order doesn't matter and this is a combination.

 

Practice questions

Question 2

Question 3