There are **107** NRICH Mathematical resources connected to **Combinations**, you may find related items under Decision Mathematics and Combinatorics.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

My coat has three buttons. How many ways can you find to do up all the buttons?

Try grouping the dominoes in the ways described. Are there any left over each time? Can you explain why?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

An environment which simulates working with Cuisenaire rods.

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Have a go at this game which involves throwing two dice and adding their totals. Where should you place your counters to be more likely to win?

Here are some rods that are different colours. How could I make a yellow rod using white and red rods?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Use these head, body and leg pieces to make Robot Monsters which are different heights.

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Can you fill in the empty boxes in the grid with the right shape and colour?

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

Find all the numbers that can be made by adding the dots on two dice.

Ben has five coins in his pocket. How much money might he have?

Noah saw 12 legs walk by into the Ark. How many creatures did he see?

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

How many different shapes can you make by putting four right- angled isosceles triangles together?

Terry and Ali are playing a game with three balls. Is it fair that Terry wins when the middle ball is red?

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

Four children were sharing a set of twenty-four butterfly cards. Are there any cards they all want? Are there any that none of them want?

Go through the maze, collecting and losing your money as you go. Which route gives you the highest return? And the lowest?

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

I start my journey in Rio de Janeiro and visit all the cities as Hamilton described, passing through Canberra before Madrid, and then returning to Rio. What route could I have taken?

When you think of spies and secret agents, you probably wouldn’t think of mathematics. Some of the most famous code breakers in history have been mathematicians.

Semaphore is a way to signal the alphabet using two flags. You might want to send a message that contains more than just letters. How many other symbols could you send using this code?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

Using 3 rods of integer lengths, none longer than 10 units and not using any rod more than once, you can measure all the lengths in whole units from 1 to 10 units. How many ways can you do this?

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?