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Chapter 6: Problem 20

a. How many distinguishable ways can the letters of the word \(M I L L I M I C R O N\) be arranged? b. How many distinguishable arrangements of the letters of \(M I L L I M I C R O N\) begin with \(M\) and end with \(N\) ? c. How many distinguishable arrangements of the letters of MILLIMICRON contain the letters \(C R\) next to each other in order and also the letters \(O N\) next to each other in order?

Short Answer

Expert verified
a) There are 831,600 distinguishable ways to arrange the letters of the word MILLIMICRON. b) There are 12,600 distinguishable arrangements of the letters of MILLIMICRON that start with M and end with N. c) There are 1,890 distinguishable arrangements of the letters of MILLIMICRON that have CR and ON next to each other in order.

Step by step solution

01

Calculate total permutations without considering repetitions

There are 12 letters in the word MILLIMICRON. Without considering repetitions, we have \(12!\) permutations.
02

Divide by the factorial of the frequency of each repeated letter

We have 4 I's, 2 L's, and 2 M's as repeated letters. We divide the total permutations from Step 1 by the factorial of the frequency of each repeated letter to get the number of distinguishable ways: \[\frac{12!}{4!2!2!}\]
03

Calculate the number of distinguishable ways

Now, utilize the formula from Step 2: \[\frac{12!}{4!2!2!} = \frac{479001600}{576} = 831600\] So, there are 831,600 distinguishable ways to arrange the letters of the word MILLIMICRON. b) Distinguishable arrangements starting with M and ending with N:
04

Fix M and N as first and last letters

We fix M as the first letter and N as the last letter, leaving us with 10 positions for the remaining 10 letters.
05

Calculate permutations considering repetitions

Similar to part a, calculate the permutations considering repetitions of I (4 times), L (2 times), and M (1 time now): \[\frac{10!}{4!2!1!}\]
06

Compute arrangements starting with M and ending with N

Use the formula from Step 2: \[\frac{10!}{4!2!1!} = \frac{3628800}{288} = 12600\] So, there are 12,600 distinguishable arrangements of the letters of MILLIMICRON that start with M and end with N. c) Distinguishable arrangements where CR and ON are next to each other:
07

Consider CR and ON as single entities

We consider CR and ON as single entities, giving us 9 characters (7 single letters, and 2 entities - CR and ON) to be arranged.
08

Calculate permutations with repetitions

Calculate the permutations considering repetitions of I (4 times), L (2 times), and M (2 times): \[\frac{9!}{4!2!2!}\]
09

Multiply by arrangements within CR and ON

Since CR and ON must stay in order, we don't need any additional permutations for them. The permutations from Step 2 remain unchanged.
10

Compute arrangements where CR and ON are next to each other

Use the formula from Step 2: \[\frac{9!}{4!2!2!} = \frac{362880}{192} = 1890\] So, there are 1,890 distinguishable arrangements of the letters of MILLIMICRON that have CR and ON next to each other in order.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial
A factorial, denoted by an exclamation mark \(n!\), is the product of all positive integers up to a given number \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials are crucial in permutations and combinatorics because they help us calculate the number of ways objects can be arranged.
  • \(0!\) is defined as 1, which is an essential base case in factorial calculations.
  • Factorials grow very rapidly, even with relatively small values of \(n\).
In the word "MILLIMICRON," there are 12 letters, so initially, the number of permutations is \(12!\), which calculates all possible arrangements if each letter was unique.
Repetitions
Repetitions occur when some elements in a set are identical, affecting how we count permutations. When elements repeat, permutations are over-counted, as identical items being swapped result in indistinguishable outcomes.
  • To adjust for these repetitions, we divide the total number of permutations by the factorials of each repeating element's frequency.
  • For "MILLIMICRON," the repeated letters are 'I' (4 times), 'L' (2 times), and 'M' (2 times). Therefore, the formula is \(\frac{12!}{4!2!2!}\).
By applying this formula, the correct number of distinguishable permutations is computed, reflecting realistic arrangements.
Combinatorics
Combinatorics is the branch of mathematics focused on counting, arrangement, and combination of elements in sets. It's integral when dealing with problems involving permutations and combinations.
  • Combinatorics helps solve problems where we count possible arrangements, like in arranging letters of a word.
  • It includes not just permutations, where order matters, but also combinations, where order does not matter.
In exercises like determining how to arrange the letters of "MILLIMICRON," combinatorial principles guide the evaluation of all possible configurations, including specific stipulations such as fixing certain elements in position or treating grouped letters as single entities.

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Most popular questions from this chapter

A coin is tossed four times. Each time the result \(H\) for heads or \(T\) for tails is recorded. An outcome of HHTT means that heads were obtained on the first two tosses and tails on the second two. Assume that heads and tails are equally likely on each toss. a. How many distinct outcomes are possible? b. What is the probability that exactly two heads occur? c. What is the probability that exactly one head occurs?

A large pile of coins consists of pennies, nickels, dimes, and quarters (at least 30 of each). a. How many different collections of 30 coins can be chosen? b. What is the probability that a collection of 30 coins chosen at random will contain at least four coins of each type?

a. How many strings of hexadecimal digits consist of from one through three digits? (Recall that hexadecimal numbers are constructed using the 16 digits \(0,1,2,3,4,5,6\), \(7,8,9, \mathrm{~A}, \mathrm{~B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F})\) b. How many strings of hexadecimal digits consist of from two through five digits?

An interesting use of the inclusion/exclusion rule is to check survey numbers for consistency. For example, suppose a public opinion polltaker reports that out of a national sample of 1,200 adults, 675 are married, 682 are from 20 to 30 years old, 684 are female, 195 are married and are from 20 to 30 years old, 467 are married females, 318 are females from 20 to 30 years old, and 165 are married females from 20 to 30 years old. Are the polltaker's figures consistent? Could they have occurred as a result of an actual sample survey?

One urn contains one blue ball (labeled \(B_{1}\) ) and three red balls (labeled \(R_{1}, R_{2}\), and \(R_{3}\) ). A second urn contains two red balls \(\left(R_{4}\right.\) and \(\left.R_{5}\right)\) and two blue balls \(\left(B_{2}\right.\) and \(\left.B_{3}\right)\). An experiment is performed in which one of the two urns is chosen at random and then two balls are randomly chosen from it, one after the other without replacement. a. Construct the possibility tree showing all possible outcomes of this experiment. b. What is the total number of outcomes of this experiment? c. What is the probability that two red balls are chosen?
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