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?
+128475 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
+2666 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.
