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Chapter 8: Problem 11

At Flo's Flower Shop, Flo wants to arrange 15 different plants on five shelves for a window display. In how many ways can she arrange them so that each shelf has at least one, but no more than four, plants?

Short Answer

Expert verified
Manually calculating this given the constraints is complicated. It requires generating viable combinations and calculating arrangements for each. Hence, it's better to develop a program or use combinatorics software to compute this.

Step by step solution

01

Initial Analysis

Flo has 15 plants and 5 shelves. She wants to arrange these plants on shelves in such a way that each of the shelves has at least one plant but no more than four. This means that each shelf needs to have 1, 2, 3 or 4 plants.
02

Determine All Possible Combinations

To find all the ways in which the plants can be arranged, determine all combinations for placing plants on shelves. These combinations can be attained by adding up all the possible ways to place 1, 2, 3 or 4 plants on a shelf across all the 5 shelves. One such combination could be (1,1,1,1,11). This combination indicates putting one plant on each of the first 4 shelves and 11 plants on the 5th shelf. Continue this way to find all the combinations where each shelf has at least one and at most four plants.
03

Calculate Number of Arrangements for Each Combination

For each of the combinations found in Step 2, find the number of arrangements of the plants possible. This is found by using the formula for permutations of multisets, which is given by \(\frac{n!}{n1!n2!n3!...nk!}\). Here, 'n' is the total number of plants (15) and 'n1', 'n2', 'n3' etc. denote the number of plants on each shelf. Add up the number of arrangements for all the combinations.
04

Total Number of Arrangements

The total number of ways for Flo to arrange the plants can be found by summing up the number of arrangements for each combination found in Step 3. This will give us the solution to the problem.

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Most popular questions from this chapter

Ms. Pezzulo teaches geometry and then biology to a class of 12 advanced students in a classroom that has only 12 desks. In how many ways can she assign the students to these desks so that (a) no student is seated at the same desk for both classes? (b) there are exactly six students each of whom occupies the same desk for both classes?

Compute \(\phi(n)\) for \(n\) equal to (a) \(51 ;\) (b) 420 ; (c) 12300 .

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