Money, Money, Money!!!

It's easy to formulate an equation. Let the number of coins 50p, £1 and £2 are $x, y, and z$ respectively.

Thus:

$0.5x+y+2z=100$

$x+2y+4z=200$ ........................................ $\boxed{1}$

Since $2y and 4z$ are always even, $x$ must be also even number.

Let:

$x=2k$ .............................................. $\boxed{2}$

$x$ in $\boxed{1}$ substituted with $\boxed{2}$

$x+2y+4z=200 \Rightarrow 2k+2y+4z=200$

$k+y+2z=100$ .............................................. $\boxed{3}$

Since $2z$ is always even, then easily find the number of making up to $100$

$k+y+2z=100$

$k+y=100-2z$

The value of $100-2z$ must be non-negative integer and $z$ must be non-negative integer. It is easy that the boundaries of $z$ are $0 \leq z \leq 50$

The number of choosing $k$ and $y$ is calculated by using combinations with repetition, since we know $z$ has the bounds, we conclude that

$\displaystyle \sum_{z=0}^{50} {100-2z+1 \choose 100-2z}$

$=\displaystyle \sum_{z=0}^{50} { 101-2z \choose 100-2z }$

$=\displaystyle \sum_{z=0}^{50} { 101-2z \choose 1 }$

$=101+99+97+...+3+1=(\frac{101+1}{2})^2=51^2=2601$

Let us try making up £100, using just 50p and £1 coins. You can do this using 0 £1 coins, 1 £1 coin, 2 £1 coins,... ...,100 £1 coins. So there are 101 ways to do this.

Now let us consider the problem with the £2 coins.

Consider the number of ways of making up £100 with 1 £2 coin. Using the same method as earlier, there is £98 left to be made up with 0-98 £1 coins so there are 99 ways. Now consider the number of ways with 2 £2 coins used. There is £96 left so using the same logic there are 97 ways. You should notice that by adding another £2 coin, the number of ways of making up £100 with that number of £2 coins decreases by 2.

So, listing the number of ways of making up £100 with 0,1, 2, ...,50 £2 coins we have an arithmetic progression: 101, 99, 97, ... , 3, 1.
Using the formula 1/2 n(a+l) to compute the sum of an arithmetic progression, we get 1/2x51x(101+1) = 1/2 x 51 x 102 = 2601

So there are 2601 ways of making up £100 using only 50p, £1 and £2 coins.