New Zealand
Level 8 - NCEA Level 3

# Mixed problem

Lesson

Identifying whether a problem is asking for a combination or permutation, comes down to the unique difference being the answer to this question:

Is the order important?

If the answer is YES - then it is a permutation (like the lock on a safe).

If a permutation then you use the notation and formula

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

Our previous notes on permutations, more permutations and probabilities with permutations are found on those links.

If the answer is NO - then it is a combination

If a combination then you use the notation and formula

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

Our previous notes on combinations, more combinations and probabilities with combinations are found on those links.

#### Worked Examples

##### QUESTION 1

When selecting people to be in a squad of rowers from a larger group, does this represent a permutation or a combination?

1. Combination

A

Permutation

B

Combination

A

Permutation

B

##### QUESTION 2

A newspaper editor is deciding which of $6$6 articles to print on the front page.

1. If she can only choose $2$2 of them for the front page, how many different selections are possible?

2. If their order on the front page matters, how many different arrangements are possible for the front page?

3. She finds an error in one of the $6$6 articles and cannot print it. How many different arrangements for the front page are now possible, given that the order of the articles on the front page still matters?

##### QUESTION 3

$4$4 letters are chosen from the word $BANNER$BANNER and are shuffled to create a new word. What is the probability that at least one of the $N$Ns appears in the new word?

### Outcomes

#### M8-3

Use permutations and combinations