Maths Problem (1 Viewer)

[chokoholic]

Maths Problems

I posted this same maths problem, in the year 10 forum, but no one seem to help me.

So I want to give it a try here. I'm really desperate!

Here is the question:-
If you have 50c, $2 and $1 coins, in how many ways can you make up $10?

The answer is 21, but it took me ages to work out because I was writing each way down.

Like this
50c x 7 + $2 x 1 + $1 x 1
50c x 6 + $2 x 2
50c x 6 + $2 x 2 + $1 x 1
and so on....

If I was to do this in an exam, it would take nearly a quarter of my exam time (because I'm a slow writer).

EDIT: I also have another problem.
Note that 1+2+3+45+6+78+9+144. In how many other ways is it possible to make a total of 144 using only 1, 2, 3, 4, 5, 6, 7, 8 and 9 in that order and addition signs?

Answer is 3, but how do you do it?

ThankYou

 

[ND]

The way i'd do it was to consider the cases:

- Only $2 coins
- Only $1 coins
- Only 50c coins
- $2 and $1 coins
- $1 and 50c coins
- $2 and 50c coins
- All coins.

Now the 1st 3 cases are just gonna be 1 each. The number of ways in the $2&$1 case is gonna be the same as the $2&50c case (cos you can only use 2 50c at a time, which = $1). For the last one, you have to have at least 2 50c, 1 $1 and 1 $2, meaning that you have $6 left to make up in any way you want. Take cases again (they'll be the same as above). I'm sure you can do all the cases except maybe the one where you need all the coins. So to make up $6 from all the coins, again, you need at least 2 50c, 1 $1 and 1 $2, meaning you have $2 left to make up in any way. Now it doesn't take long to count the number of ways you can make $2 from those coins.

This is definately a difficult yr10 question. (but i have a feeling that someone's gonna come up with a 3 line answer or something).

edit: and to get the total number of ways, sum each case.

 

[CM_Tutor]

Are you sure that the answer is 21? I only see 20, and if it is only 20, then there is a very easy way to do it.

 

[ND]

Umm are you sure it's only 20? I get way more than 20:
- Only $2 coins - 1 way
- Only $1 coins - 1 way
- Only 50c coins - 1 way
- $2 and $1 coins - 4 ways
- $1 and 50c coins - 8 ways
- $2 and 50c coins - 4 ways

There's already 19 there, and i haven't even considered the combinations with all 3 coins...

 

[CM_Tutor]

You're right, and my very easy way is clearly wrong. Oops.

 

[chokoholic]

So what's the way?