How many ways to choose, how many numbers can be made, handshakes, counting paths, making change with coins and picking marbles to be sure (the pigeonhole principle). Children learn to list with a table or a tree.
How many ways to choose
Solving tip: Try matching each shirt with every pair of pants.
Bunny has 2 shirts and 3 pairs of pants. How many ways can Bunny pick 1 shirt and 1 pair of pants to wear?
Show solution
Answer: 6
How to think: Try matching each shirt with every pair of pants.
Solution: Each shirt goes with 3 pairs of pants, and there are 2 shirts: 3 plus 3 equals 6 ways.
How many numbers can you make
Solving tip: Choose the first digit first, then the next digits.
Using the digits 8, 4, 7, how many numbers with 2 different digits can you make?
Show solution
Answer: 6
How to think: Choose the first digit first, then the next digits.
Solution: The first place has 3 choices, and for each one the next place has 2 choices: 2 plus 2 plus 2 equals 6 numbers.
Which number is the k-th largest
Solving tip: Compare the first digits first. If the first digits are the same, compare the next digits.
Put the numbers 28, 74, 51, 82 in order from biggest to smallest. Which number is in place 2?
Show solution
Answer: 74
How to think: Compare the first digits first. If the first digits are the same, compare the next digits.
Solution: From biggest to smallest: 82, 74, 51, 28. The number in place 2 is 74.
Ways to share sweets
Solving tip: Count by how many candies the fox gets: 0, 1, 2, and so on.
Share 8 candies between the fox and the frog. One of them may get none. How many ways are there?
Show solution
Answer: 9
How to think: Count by how many candies the fox gets: 0, 1, 2, and so on.
Solution: The fox can get from 0 to 8 candies, and each number is one way: 9 ways.
Ways to line up
Solving tip: Choose who stands first, then who stands second, and so on to the last one.
There are three friends: fox, bear cub, piglet. They stand in a line. How many different ways can they line up?
Show solution
Answer: 6
How to think: Choose who stands first, then who stands second, and so on to the last one.
Solution: For each friend at the front, the other two can stand 2 ways. 3 friends can be at the front: 2 plus 2 plus 2 equals 6 ways.
Handshakes
Solving tip: The first team plays the 4 other teams.
5 teams play, and every two teams play each other exactly once. How many games are there in all?
Show solution
Answer: 10
How to think: The first team plays the 4 other teams.
Solution: Add up each team's new games: 4 plus 3 plus 2 plus 1 equals 10 games.
Counting paths
Solving tip: Points on the bottom edge and on the left edge have only 1 way to get there.
Follow the lines. Each step can only go right or up. How many ways are there to go from A to B?
Show solution
Answer: 6
How to think: Points on the bottom edge and on the left edge have only 1 way to get there.
Solution: Write the number of ways at each point: each point is the point on its left plus the point below. There are 6 ways to reach B.
Coin combinations
Solving tip: Start by using as many coins worth 2 as you can.
How many ways can you make 7 coins using coins worth 1, 2? You can take any number of each kind, and the order doesn't matter.
Show solution
Answer: 4
How to think: Start by using as many coins worth 2 as you can.
Solution: List them one by one, starting with the most coins worth 2, without missing any: there are 4 ways.
Picking marbles to be sure
Solving tip: Think of the unluckiest case: the marbles you take out are never the ones you want.
A box has 3 red marbles and 4 blue marbles. Without looking, what is the smallest number of marbles you must take out to be sure to get 1 red marble?
Show solution
Answer: 5
How to think: Think of the unluckiest case: the marbles you take out are never the ones you want.
Solution: The unluckiest case: you take out all 4 blue marbles first, then the next one is sure to be red: 4 plus 1 equals 5 marbles.
Making even and odd numbers
Solving tip: An odd number ends in 1, 3, 5, 7 or 9. Choose the ones digit first.
Using the digits 5, 6, 2, how many odd numbers with 2 different digits can you make?
Show solution
Answer: 2
How to think: An odd number ends in 1, 3, 5, 7 or 9. Choose the ones digit first.
Solution: An odd number ends in 1, 3, 5, 7 or 9. The numbers are: 25, 65. That makes 2 numbers.
Choosing and giving prizes
Solving tip: Pair the first thing with each other thing, then the second thing with the ones not paired yet.
There are 4 different things in the picture. You choose 2 of them. How many ways can you choose?
Show solution
Answer: 6
How to think: Pair the first thing with each other thing, then the second thing with the ones not paired yet.
Solution: The first thing pairs with 3 things. Each next one makes one fewer new pair: 3 plus 2 plus 1 equals 6 ways.
Paying with coins
Solving tip: Try paying with one coin, then two coins, then three. Write down each amount.
You have coins marked 1, 2, 10. Without getting change, how many different amounts can you pay exactly?
Show solution
Answer: 7
How to think: Try paying with one coin, then two coins, then three. Write down each amount.
Solution: The amounts you can pay are 1, 2, 3, 10, 11, 12, 13. That makes 7 amounts.
Sure even when unlucky
Solving tip: Think of the unluckiest case: the marbles you take out are never the ones you want.
A box has 6 red marbles, 5 blue marbles and 2 yellow marbles. Without looking, at least how many marbles must you take out to be sure to have all 3 colours?
Show solution
Answer: 12
How to think: Think of the unluckiest case: the marbles you take out are never the ones you want.
Solution: The unluckiest case is taking all of the two biggest colours first: 6 plus 5 equals 11 marbles, and one colour is still missing. The next marble must be the last colour: 11 plus 1 equals 12.
Keep practising these types in the Karotto app
In the app every problem type above is a lesson with the question read aloud, three levels of hints when your child is stuck and a step-by-step solution. Problems are generated fresh, so every attempt is a new set.