The fundamental principle of counting
Aoife is buying lunch in a café in Galway. There are 3 sandwiches and 2 drinks on the menu. How many different lunches can she choose from? You could write them all out, but there is a much faster way, and it still works when the lists are far too long to write out.
Listing the choices with a tree diagram
A tree diagram shows every possible outcome. Each choice is a set of branches. Follow a path from left to right and you build one complete lunch.
There are 3 first branches, and each one splits into 2. That gives $3 \times 2 = 6$ lunches. Notice that we are adding up 2 + 2 + 2, which is the same as $3 \times 2$.
A café in Kilkenny offers $3$ types of sandwich and $4$ types of drink. How many different meals consisting of one sandwich and one drink are possible?
A café in Kilkenny offers $3$ types of sandwich and $4$ types of drink. How many different meals consisting of one sandwich and one drink are possible?
A school jumper comes in $5$ sizes and $2$ colours. How many different jumpers are available?
A school jumper comes in $5$ sizes and $2$ colours. How many different jumpers are available?
The fundamental principle of counting
It works for any number of choices: just multiply the number of options at each stage.
A coin is tossed and then a standard six-sided die is rolled. How many different outcomes are possible?
A coin is tossed and then a standard six-sided die is rolled. How many different outcomes are possible?
A locker code uses $1$ letter from A–F followed by $2$ digits from $0$–$9$ (digits may repeat). a) How many codes are possible? b) How many codes are possible if the…
A locker code uses $1$ letter from A–F followed by $2$ digits from $0$–$9$ (digits may repeat).
- a) How many codes are possible?
- b) How many codes are possible if the digits must be different?
When choices cannot be repeated
Five runners finish a race. How many ways can the gold, silver and bronze medals be awarded? A runner cannot win two medals, so the number of options drops by one after each choice.
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Gold: 5 runners to choose from.
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Silver: 4 left.
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Bronze: 3 left.
The total is $5 \times 4 \times 3 = 60$ ways.
Fill the restricted place first
Sometimes a place has a condition, such as "the number must be even". Deal with that place first, then fill the others.
Five runners, Aoife, Brian, Ciara, Dara and Éabha, finish a race with no ties. a) How many ways can the first three places be filled? b) In how many of these is Ciara…
Five runners, Aoife, Brian, Ciara, Dara and Éabha, finish a race with no ties.
- a) How many ways can the first three places be filled?
- b) In how many of these is Ciara first?
How many three-digit numbers can be made from the digits $1, 2, 3, 4, 5$ if a) digits can be repeated? b) digits cannot be repeated? c) digits cannot be repeated and the…
How many three-digit numbers can be made from the digits $1, 2, 3, 4, 5$ if
- a) digits can be repeated?
- b) digits cannot be repeated?
- c) digits cannot be repeated and the number is even?
Conditions, restrictions and checking
Sometimes one particular combination is not allowed. Count all the possibilities, then subtract the ones that are not allowed.
Another useful idea is to split the cases. For example, to count the committees that include a particular student, first decide which role she gets, then fill the other roles.
Seán has $4$ T-shirts, $3$ pairs of shorts and $2$ pairs of runners. a) How many different outfits can he make? b) How many if he refuses to wear his red T-shirt with…
Seán has $4$ T-shirts, $3$ pairs of shorts and $2$ pairs of runners.
- a) How many different outfits can he make?
- b) How many if he refuses to wear his red T-shirt with his green shorts?
A car registration plate in a game has $2$ letters from A–Z followed by $3$ digits from $0$–$9$. a) How many plates are possible if repetition is allowed? b) How many…
A car registration plate in a game has $2$ letters from A–Z followed by $3$ digits from $0$–$9$.
- a) How many plates are possible if repetition is allowed?
- b) How many have all five characters different?
- c) Explain why your answer to b) is smaller than a).
A committee of a chairperson, a secretary and a treasurer is chosen from $8$ students, and no student may hold two roles. a) How many different committees are possible?…
A committee of a chairperson, a secretary and a treasurer is chosen from $8$ students, and no student may hold two roles.
- a) How many different committees are possible?
- b) If Niamh must be chosen for one of the three roles, how many committees are possible?
- c) Show that the number of committees without Niamh plus the number with her equals your answer to a).
Back to the start: Aoife's café lunch can be chosen in $3 \times 2 = 6$ ways, just as the tree showed.
Summary
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A tree diagram lists every outcome: follow each path from left to right.
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Fundamental principle of counting: multiply the number of ways for each choice.
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Do not add the numbers of options; multiply them.
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If repeats are not allowed, the number of options drops by one at each stage.
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Fill the place with a condition first (such as an even number).
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For a banned combination, subtract it from the total, and check by splitting into cases.