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# Counting

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36
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Kurt is creating a bouquet of flowers for his mother. He has a selection of roses, lilies, orchids and violets from which to create a bouquet of a dozen flowers. Kurt's mother loves orchids, so his bouquet will have at least eight orchids. How many different groups of a dozen flowers can Kurt use to make a bouquet?

Jul 13, 2022

#1
+124525
+1

Suppose we have 12 members of a set

We have 8 orchids in  the set by default

Of the other 4  flowers in  the set (making a  dozen in total)  we have 4 ways to choose a flower in  each of the remaining 4 positions

So.....total number of possible boquets (sets)  = 4 * 4 * 4 * 4  =    256

Jul 21, 2022
#2
+2283
+1

Note that we use stars and bars for this method to prevent overcounting.

We have 4 stars and 3 bars to sort the 4 remaining flowers, and the number of orderings is how many ways we can order the stars and bars, making for \({7 \choose 3} = \color{brown}\boxed{35}\) options.

Jul 21, 2022
edited by BuilderBoi  Jul 21, 2022