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Problem 31

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Coin Sums

In the United Kingdom the currency is made up of pound ($\mathit{\text{£}}$) and pence ($p$). There are eight coins in general circulation:
$$1p,2p,5p,10p,20p,50p,\mathit{\text{£}}1(100p),\mathit{\text{£}}2(200p).$$

It is possible to make $\mathit{\text{£}}2$ in the following way:
$$1\times \mathit{\text{£}}1+1\times 50p+2\times 20p+1\times 5p+1\times 2p+3\times 1p$$

How many different ways can $\mathit{\text{£}}2$ be made using any number of coins?

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硬币求和

英国的货币单位分为英镑（$\mathit{\text{£}}$）和便士（$p$）。目前流通的硬币一共有八种面值：
$$1p,2p,5p,10p,20p,50p,\mathit{\text{£}}1(100p),\mathit{\text{£}}2(200p).$$

想要凑出$\mathit{\text{£}}2$，其中一种做法是：
$$1\times \mathit{\text{£}}1+1\times 50p+2\times 20p+1\times 5p+1\times 2p+3\times 1p$$

不限制硬币数量，凑出$\mathit{\text{£}}2$有多少种不同的做法？

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