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Magazine Chapter 5: Problem 7
List all permutations of five objects \(a, b, c, d,\) and \(e\) taken three at a time without replacement.
Short Answer
Expert verified
There are 60 unique permutations of three objects taken from a, b, c, d, and e.
Step by step solution
01
- Understand permutations
A permutation is an arrangement of objects in a specific order. Since the problem asks for permutations of five objects taken three at a time without replacement, we will list all unique sequences of length three formed from the given objects.
02
- Identify the objects
The objects given are: a, b, c, d, and e.
03
- Generate permutations
We'll list permutations by arranging the objects in groups of three without repeating any object in a single permutation.
04
- List permutations starting with 'a'
Permutations starting with 'a': abc, abd, abe, acb, acd, ace, adb, adc, ade, aeb, aec, aed.
05
- List permutations starting with 'b'
Permutations starting with 'b': bac, bad, bae, bca, bcd, bce, bda, bdc, bde, bea, bec, bed.
06
- List permutations starting with 'c'
Permutations starting with 'c': cab, cad, cae, cba, cbd, cbe, cda, cdb, cde, cea, ceb, ced.
07
- List permutations starting with 'd'
Permutations starting with 'd': dab, dac, dae, dba, dbc, dbe, dca, dcb, dce, dea, deb, dec.
08
- List permutations starting with 'e'
Permutations starting with 'e': eab, eac, ead, eba, ebc, ebd, eca, ecb, ecd, eda, edb, edc.
09
- Verify and count the permutations
Verify that each permutation listed is unique and count the total permutations. There should be 60 permutations: 5 choices for the first spot, 4 choices for the second spot, and 3 choices for the third spot, totaling 5 脳 4 脳 3 = 60.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Permutations
A permutation refers to the arrangement of objects in a specific sequence.
It's important to remember that the order of these objects matters in permutations.
For instance, in the problem where we have five objects (a, b, c, d, and e) and we need to take three at a time without replacement, each different sequence counts as a unique permutation.
For example, abc is different from acb, as the order of the letters is different.
It's important to remember that the order of these objects matters in permutations.
For instance, in the problem where we have five objects (a, b, c, d, and e) and we need to take three at a time without replacement, each different sequence counts as a unique permutation.
For example, abc is different from acb, as the order of the letters is different.
Combinatorics
Combinatorics is a branch of mathematics dealing with combinations and permutations of objects.
It includes solving problems related to counting, arranging, and analyzing discrete structures.
The problem provided is a typical combinatorial challenge.
We need to find and list all the unique sequences made up of three out of five objects.
This exercise helps you understand how to systematically approach such problems and utilize methods to list all possible permutations.
It includes solving problems related to counting, arranging, and analyzing discrete structures.
The problem provided is a typical combinatorial challenge.
We need to find and list all the unique sequences made up of three out of five objects.
This exercise helps you understand how to systematically approach such problems and utilize methods to list all possible permutations.
Counting Principles
Counting principles are fundamental rules in combinatorics used to count possible arrangements of objects.
In the given problem, the principle of counting permutations is applied.
Here鈥檚 how: We have 5 choices for the first spot, after choosing the first object, we have 4 remaining choices for the second spot, and 3 choices left for the third spot.
Therefore, the total number of permutations can be calculated as:
\[ 5 \times 4 \times 3 = 60 \]
This step-by-step approach helps in breaking down complex counting tasks into manageable parts.
In the given problem, the principle of counting permutations is applied.
Here鈥檚 how: We have 5 choices for the first spot, after choosing the first object, we have 4 remaining choices for the second spot, and 3 choices left for the third spot.
Therefore, the total number of permutations can be calculated as:
\[ 5 \times 4 \times 3 = 60 \]
This step-by-step approach helps in breaking down complex counting tasks into manageable parts.
Unique Sequences
Unique sequences in permutations mean that no repetition of objects is allowed in each arrangement.
This means each sequence of the objects chosen is different from the others.
For instance, the sequences abc, acb, and bac are unique because even though they consist of the same letters, their orders are distinct.
In the given problem, after listing all permutations, it is essential to verify that each sequence is unique.
This is part of ensuring you have correctly applied the permutation principles.
This means each sequence of the objects chosen is different from the others.
For instance, the sequences abc, acb, and bac are unique because even though they consist of the same letters, their orders are distinct.
In the given problem, after listing all permutations, it is essential to verify that each sequence is unique.
This is part of ensuring you have correctly applied the permutation principles.
