Combining Coins

Charlie has 3 coins in his pocket that add up to 32p:
20p 10p 2p.

Alison has 7 coins in her pocket that also add up to 32p:
10p, 5p, 5p, 5p, 5p, 1p, 1p.

Matt has 32 coins in his pocket that also add up to 32p. I'm not going to list them, you should be able to work out what he has!

Is there a way of making 32p that uses 4 coins? or 5 coins? or ANY number of coins between 3 and 32?

What about other totals? Is it always possible to make a total using every number of coins between the minimum and the maximum?

******
"I'm thinking of a value that I can make in 5 different ways using just 1p and 2p coins."
"Well it could be 8p, because I can make that using:
4x2
3x2, 2x1
2x2, 4x1
1x2, 6x1
0x2, 8x1

"Oh yes, that does work, doesn't it! I was actually thinking of 9p though..."

How many ways are there of making different amounts using just 1p and 2p coins?
Which totals have the same number of ways?
Is there a quick way of working out the number of ways of making an amount with just 1p and 2p coins?

What if we just had 1p and 3p coins instead?
Or just 1p and 4p coins?

*****

Emmy has been working out different ways to make 12p using 1p, 2p, 5p and 10p coins.

So far, this is what she has done:

With a 10p I need 2p more so there's 2 ways of doing that.
With two 5ps, I also need 2p more so there's 2 ways of doing that...

10p 5p 2p 1p Number of ways
         
1 of these none of these     2
none of these 2 of these     2
none of these 1 of these      
... ...      
... ...      

Can you help Emmy to finish off? You should find there are 15 ways altogether!

Can you describe a method for working out how many ways there are to make any amound using 10p, 5p, 2p and 1p coins?
What if you also had 20p, 50p coins?
It might get very time-consuming even if you have a good method!
Could you write a computer program to help?

*****



Want to share what you've done? Got any interesting questions or ideas? Send us your thoughts.