# Clever counting » Diagram/Grid

## Example 1: Tree diagram

How many possibilities are there to throw at least 16 with three dice?

Answer:

Start with choosing the largest possibility.

You can make a tree diagram to do this systematically.

There are 10 possibilities to throw at least 16 with three dice.

## Example 2: Grid/table

How many possibilities are there to throw at least 9 with two dice?

Answer:

Because we are dealing with two dice, you can use a scheme to easily show the different possibilities of throwing with two dice.

There are 6 possibilities to throw at least 9 with 2 dice.

## Example 3: Road diagram

Use the digits 2, 3, 4, 5 en 6 to make a 4-digit number, in which you may use the digits more than once.

How many numbers greater than 5400 can be made?

Answer:

Now we can use a 'road diagram'.

If the first digit is a 5, the second digit must be a 4, 5 or 6 and for the third and fourth digit there are no limitations.

If the first digit is a 6, there are no limitations for the second, third and fourth digit.

3 × 5 × 5 = 75 | |

5 × 5 × 5 = 125 |

There are 75 + 125 = 200 numbers that can be made greater than 5400.

## Example 4: Road diagram

Use the digits 2, 3, 4, 5 en 6 to make a 4-digit number, in which you may use the digits only once.

How many numbers greater than 5400 can be made?

Answer:

If we start with a 5, the second digit can only be a 4 or 6.

If the second digit is a 4, there are only three possibilities for the third digit (we already used two). The same applies for a 6.

In both cases you will have two possibilities left for the fourth digit.

If we start with a 6, there are four possibilities for the second digit, three possibilities for the third digit and two possibilities for the fourth digit.

2 × 3 × 2 = 12 | |

4 × 3 × 2 = 24 |

There are 12 + 24 = 36 numbers that can be made greater than 5400.