 # Use Multiplicative Counting

In this worksheet, students will use the method of multiplicative counting to calculate the total number of possible outcomes of linked events, taking into account dependent events (when the result in an earlier choice affects the later choices) or repeated outcomes. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Probability

Curriculum subtopic:   Structure and Calculation, Combined Events and Probability Diagrams

Difficulty level:   ### QUESTION 1 of 10

What on earth is multiplicative counting, you ask?

Multiplicative counting is a technique in maths that allows us to find the total number of combinations, when presented with a number of different options.

e.g. If you went to a deli and they had 6 types of bread, 4 types of filling and 9 sauces, how many different sandwiches could you make (using one type of bread, one filling and one sauce for each sandwich)?

Obviously, we could list all the combinations out and count them individually, but this would take forever!

The trick here is to multiply the options:

9 x 4 x 9 = 324 options

This systematic approach works no matter how large the numbers get or how many individual events are linked.

e.g. A pin number has 4 digits, how many different options are there for PIN numbers?

The four digits have 10 options (0 - 9), so there are:

10 x 10 x 10 x 10 = 10000 different options

That's why it's so hard to guess someone's pin number!

Let's extend this now...

A pin number has 4 digits, how many different options are there for PIN numbers if a user cannot re-use a number?

The first digit can be chosen from 10 options.

The second digit can be chosen from 9 options.

The third can be chosen from 8 options.

Finally, the fourth can be chosen from only 7 options.

So all together in this circumstance, a user has:

10 x 9 x 8 x 7 = 5040 options

e.g. Mahlia put the five vowels from the alphabet into a bag and pulls out two. How many different combinations could she make?

If you follow our work from the previous questions, you would think there should be 20 options (5 x 4).

But there's actually only 10!

This is sometimes the case and occurs when the order in which events occur does not matter.

In this question, Mahlia pulls out two vowels so, for example, AE is the same outcome as EA and OU is the same as UO.

When we get this situation, we must to divide our answer by 2, as half of the outcomes are repeated.

In this activity, you will use the method of multiplicative counting described above to calculate the total number of possible outcomes of linked events, taking into account dependent events (when the result in an earlier choice affects the later choices) or repeated outcomes.

There are 12 boys and 14 girls in a class.

The class' teacher wants one boy and one girl to run an errand.

How many ways can the teacher pick one boy and one girl?

There are 12 boys and 14 girls in a class.

The class' teacher wants two girls to run an errand.

How many ways can the teacher pick two girls?

182

91

How many different ways can 8 books be arranged on a shelf?

2 positions are available on a school council.

If 10 students apply, how many different combinations of pairings could there be?

A bike comes in 5 different frame sizes with 3 different sizes of wheel.

How many different combination of bikes are there?

8

15

A password for a computer can contain only lowercase letters and numbers.

The password has to be 4 characters long.

How many different password combinations are there?

Let's consider the same scenario again but, this time, imagine that using the same numbers or letters is not allowed.

A password for a computer can contain only lowercase letters and numbers.

The password has to be 4 characters long.

How many different password combinations are there?

Complete the sentence below to summarise an exception to consider when using multiplicative counting.

Match each situation below with its total number of possible outcomes.

## Column B

Picking 3 single digit numbers in a row (can't pic...
1024
Rolling 4 dice in a row
1296
Picking 3 consonants at random (can repeat)
9261
Throwing 10 coins at once
720

10 people are in a room and each one shakes the hand of everyone else exactly once.

How many handshakes occur for each person?

• Question 1

There are 12 boys and 14 girls in a class.

The class' teacher wants one boy and one girl to run an errand.

How many ways can the teacher pick one boy and one girl?

168
EDDIE SAYS
There are 12 options for the boy and 14 for the girls. The events here are not linked, as the teacher's choice of a girl will not affect the total number of boys they can then choose from after. To calculate the total number of possible outcomes, we need to work out: 14 x 12 = 168
• Question 2

There are 12 boys and 14 girls in a class.

The class' teacher wants two girls to run an errand.

How many ways can the teacher pick two girls?

91
EDDIE SAYS
The important fact to recognise here is that the teacher's first choice makes an impact on the total number of options available for the second choice, as the same girl cannot be chosen twice. So there are 14 options for the first girl and 13 for the second girl. However, does the order in which the girls are chosen have any effect on the outcome? No, choosing Beth and Rubi is the same as choosing Rubi and Beth (for example). Therefore, this is one of the situations where we would have repeated combinations, so we need to remember to divide by 2. To find the total number of possible outcomes, we must calculate: (14 x 13) ÷ 2 = 91
• Question 3

How many different ways can 8 books be arranged on a shelf?

40320
40,320
EDDIE SAYS
For the first book, there are 8 positions to choose from. For the second, there is 7. For the third, there is 6 and so on. So to find all the possible calculations, we need to work out: 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320 Wow, that's quite a few!
• Question 4

2 positions are available on a school council.

If 10 students apply, how many different combinations of pairings could there be?

45
EDDIE SAYS
There are 10 options for the first place on the council, but only 9 for the second as the same student couldn't have both positions. However, the order in which the students are chosen has no effect on the outcome. If Ben and Katana got the positions for example, Katana and Ben would also be the same result. So we need to remember to divide our total by 2 as repeated results are possible. To calculate the overall number we need to work out: (10 x 9) ÷ 2 = 45
• Question 5

A bike comes in 5 different frame sizes with 3 different sizes of wheel.

How many different combination of bikes are there?

15
EDDIE SAYS
The common mistake to avoid is to calculate that 5 options plus 3 options gives 8 options overall. Don't forget that this question actually means there are 3 options for each of the first 5 options, so you must multiply. To calculate the total number of possible outcomes, we need to work out: 5 x 3 = 15
• Question 6

A password for a computer can contain only lowercase letters and numbers.

The password has to be 4 characters long.

How many different password combinations are there?

1679616
1,679,616
EDDIE SAYS
The question does not say that reusing the same numbers or letters is not allowed, so we can assume that the total number of outcomes remains the same for each of the 4 characters. There are 26 letters and 10 numbers to choose from, so each character has 36 different options. So to calculate all the possible password options, we need to work out: 36 x 36 x 36 x 36 = 1679616
• Question 7

Let's consider the same scenario again but, this time, imagine that using the same numbers or letters is not allowed.

A password for a computer can contain only lowercase letters and numbers.

The password has to be 4 characters long.

How many different password combinations are there?

1413720
1,413,720
EDDIE SAYS
There are 26 letters and 10 numbers. so each digit has 36 different options. However, this time the question says we cannot re-use the same characters, so the second space has 35 options, the third has 34 options, and the fourth has 33. So as the password has four digits, our calculation would be: 36 x 35 x 34 x 33 = 1,413,720 Does that make sense? Let's consider some more scenarios now.
• Question 8

Complete the sentence below to summarise an exception to consider when using multiplicative counting.

EDDIE SAYS
Remember that when the order of events is not important, there will be repeated answers. This means that the outcome a, b and b, a should be counted as the same. In this scenario, we need to divide the total number of outcomes by 2.
• Question 9

Match each situation below with its total number of possible outcomes.

## Column B

Picking 3 single digit numbers in...
720
Rolling 4 dice in a row
1296
Picking 3 consonants at random (c...
9261
Throwing 10 coins at once
1024
EDDIE SAYS
There is a lot going on here, but all you have to do is think about each scenario separately and calculate the totals. If we can have the same option for all events (this would have to be true for rolling dice,) we can multiply the same number of options for each one: 6 x 6 x 6 x 6 = 1296 If we can't reuse a value (e.g. the first scenario), we need to subtract one from the number of options each time: 10 x 9 x 8 = 720 Can you use this logic to work out the other two matches?
• Question 10

10 people are in a room and each one shakes the hand of everyone else exactly once.

How many handshakes occur for each person?

45
EDDIE SAYS
There are two ways of thinking about this: 1) The first person shakes 9 hands, then 8, then 7 and so on. This gives 45 handshakes in total for each person. 2) There are 10 x 9 handshakes. As handshakes cannot be repeated, we need to halve this to reach 45 handshakes. Great work, you've completed another activity! How about attempting another one so you feel super confident?
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