Permutations of Indistinguishable ObjectsDate: 09/21/2000 at 18:12:33 From: Candice Subject: Permutations of Indistinguishable Objects and Multiplication Principle Hi Dr. Math, I'm hoping you can help me out with two questions. 1. A forester selects four pink and four white dogwoods. The trees are to be planted in row. If a tree is distinguished by color only, in how many ways can the eight dogwoods be planted? How many of these arrangements have at least two trees of the same color side by side? I found the number of possibilities to be 576 (4x3x2x1x4x3x2x1), but I don't know how to find the second answer. 2. A study is being designed to investigate the effect of polymer type, temperature, radiation dose, radiation dose rate, and pH on the ability to remove trace quantities of benzene from water. There are 2 polymer types (A and B), 3 temperatures (high, medium, low), 3 radiation doses, 3 radiation dose rates and 3 pH levels (acidic, basic, neutral). a) If each experimental condition is to replicated 5 times, how many experimental runs must be made? b) How many runs are made with polymer B at high or medium temperature and low pH? For a) I got 810 - number of possible conditions x 5. For b) I got 1x2x1x3x3 = 18. Is this correct? Thank you in advance. Date: 09/21/2000 at 19:18:34 From: Doctor Anthony Subject: Re: Permutations of Indistinguishable Objects and Multiplication Principle >If a tree is distinguished by color only, in how >many ways can the eight dogwoods be planted? The number of arrangements is 8!/[4!4!] = C(8,4) = 70 >How many of these >arrangements have at least two trees of the same color side by side? It is best to find the number of ways with no two of the same colour side by side, and subtract this number from 70. Suppose we place the 4 white trees in a row with spaces between them and then place the pink trees as shown: P * P * P * P * or * P * P * P * P and these are the only two ways that we have no two of the same colour side by side. So the number of ways with at least two side by side is 68. >2a) If each experimental condition is to replicated 5 times how many >experimental runs must be made? 5 x 2 x 3 x 3 x 3 x 3 = 810 >2b) How many runs are made with polymer B at high or medium >temperature and low pH? 5 x 1 x 2 x 3 x 3 x 1 = 90 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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