There are **137** 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?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

How many six digit numbers are there which DO NOT contain a 5?

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?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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?

Which of these games would you play to give yourself the best possible chance of winning a prize?

Two brothers belong to a club with 10 members. Four are selected for a match. Find the probability that both brothers are selected.

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?

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

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.

Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route?

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?

From the atomic masses recorded in a mass spectrometry analysis can you deduce the possible form of these compounds?

Can all but one square of an 8 by 8 Chessboard be covered by Trominoes?

A man has 5 coins in his pocket. Given the clues, can you work out what the coins are?

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

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?

Using only the red and white rods, how many different ways are there to make up the other rods?

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?