# Selecting cakes if we require at the very least among each type

I require $n$ cakes for an event. I most likely to the cake store and also there are $k$ various sort of cake. For selection, I would certainly such as to access the very least among each cake.

The amount of means can I do this?

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2019-05-07 02:53:46
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Similar to the stars and bars technique, take into consideration the n cakes as a row of n celebrities *. As opposed to permuting them with k - 1 bars | (which permits 2 bars alongside each various other, offering 0 of a type, position the k - 1 bars (required to divide the n stars right into k kinds) right into the n - 1 rooms in between the celebrities, permitting at the majority of one bar per room. The variety of means to do this is ${n-1 \choose k-1}$.
At the same time, given that you require among each type, there are just n - k cakes for which you are picking kinds. Making use of the stars and bars technique, there are ${(n-k)+k-1 \choose k-1} = {n-1 \choose k-1}$ means to do it.