Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in
every possible order to give 5 digit numbers. Find the sum of the
In how many ways can a pound (value 100 pence) be changed into
some combination of 1, 2, 5, 10, 20 and 50 pence coins? Remember,
the aim is not just to get the answer but to find a good method and
to explain it well.
There are more than 4000 possibilities so when you try this
question you will find that counting all possibilities is too
tedious unless you have a good system to reduce the work and a good
notation to record work in progress. If you are going to get the
answer you will need to find a good method which you can explain
Here is one method you might like to follow. Using a spreadsheet
saves work but it is still easy to do without one.
Use the notation 100(1,2,5,10,20,50) for the number of
combinations of the listed smaller coins which make up one pound
and similarly for smaller amounts, for example 30(1,2,5) is the
number of combinations of 1p, 2p and 5p coins which make up
Step 1 Show that the number of ways of changing X pence into 1p
and 2p coins is (X/2 + 1) when X is even and (X + 1)/2 when X is
odd. Now fill in column A in the table below.
Step 2 Fill in column B in the table using the results in column
A and using the earlier results as you work your way down the
column. For example we can make up 10 pence using no 5p coins, or
one 5p coin or two 5p coins, hence:
10(1,2,5) = 10(1,2) + 5(,1,2) + 1 = 6 + 3 + 1 = 10
and similarly to make up 20 pence we use zero, one, two, three
or four 5p coins giving:
20(1,2,5) = 20(1,2) + 15(1,2) + 10(1,2) + 5(1,2) + 1 = 11 + 8 +
6 + 3 + 1 = 29
Step 3 Fill in column C where, for example, corresponding to
zero, one, two and three 10p coins we get:
30(1,2,5,10) = 30(1,2,5) + 20(1,2,5) + 10(1,2,5) + 1 = 58 + 29 +
10 + 1 = 98
Step 4 Now you should be able to continue in this manner to fill
in the whole table and to get the answer in the bottom right hand
Table showing the numbers of combinations of smaller
coins to make up the amounts shown:
There are other ways to do this and you might like to find a
different method of your own, perhaps writing a computer program to
find the result, do let us know. It would be cool to publish
several different methods.