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Chef's bakery sells $N$ types of cakes, numbered from $1$ to $N$. The $i$-th type of cake has a weight of $i$ units. The bakery just received an order: a customer would like to buy **exactly** $K$ cakes with a **total weight** that equals $S$ units. There are no other restrictions - that is, the order can include any number of cakes of each type, as long as there are exactly $K$ cakes with th

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