Card 1 of 20
EasyStudy card 1
Prompt
What does the Fundamental Counting Principle tell you to do?
Answer
Multiply the number of legal choices at each sequential stage.
Example
Three choices followed by four choices produce \(3\times4=12\) outcomes.
MathChapter 9: Categorical Data and Probability
Flashcards
Permutations and Combinations Flashcards
Twenty concise cards on stage counting, factorials, ordered arrangements, unordered selections, and method choice.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What is \(n!\) in this chapter's positive whole-number setting?
Answer
The product of every positive whole number from \(n\) down to \(1\).
Formula
\[n!=n(n-1)\cdots2\cdot1\]
Card 3 of 20
EasyStudy card 3
Prompt
Evaluate \(5!\).
Answer
\(120\)
Example
\(5\cdot4\cdot3\cdot2\cdot1=120\)
Card 4 of 20
EasyStudy card 4
Prompt
What is a permutation?
Answer
An arrangement or selection in which order, rank, position, or role matters.
Card 5 of 20
EasyStudy card 5
Prompt
What is a combination?
Answer
An unordered selection in which only membership matters.
Card 6 of 20
EasyStudy card 6
Prompt
What question distinguishes permutations from combinations?
Answer
Would changing the order or assigned roles create a different outcome?
Card 7 of 20
MediumStudy card 7
Prompt
What is the permutation formula?
Answer
\({}_nP_r=\frac{n!}{(n-r)!}\)
Formula
\[{}_nP_r=\frac{n!}{(n-r)!}\]
Card 8 of 20
MediumStudy card 8
Prompt
What is the combination formula?
Answer
\({}_nC_r=\frac{n!}{r!(n-r)!}\)
Formula
\[{}_nC_r=\frac{n!}{r!(n-r)!}\]
Card 9 of 20
EasyStudy card 9
Prompt
In \({}_nP_r\) or \({}_nC_r\), what do \(n\) and \(r\) mean?
Answer
\(n\) is the number available; \(r\) is the number arranged or selected.
Card 10 of 20
MediumStudy card 10
Prompt
Why does the combination formula divide by \(r!\)?
Answer
To remove the \(r!\) different orders that represent the same selected group.
Card 11 of 20
EasyStudy card 11
Prompt
A captain and recorder are chosen from a club. Permutation or combination?
Answer
Permutation, because the two roles are distinct.
Card 12 of 20
EasyStudy card 12
Prompt
Four members are selected for an unlabeled committee. Permutation or combination?
Answer
Combination, because rearranging the same four members changes nothing.
Card 13 of 20
MediumStudy card 13
Prompt
How many ways can all \(n\) distinct objects be arranged?
Answer
\(n!\)
Formula
\[{}_nP_n=n!\]
Card 14 of 20
MediumStudy card 14
Prompt
What happens to slot counts when repetition is forbidden?
Answer
The number of available choices decreases after each selection.
Example
Six choices for the first slot, then five, then four.
Card 15 of 20
MediumStudy card 15
Prompt
What is the fastest way to simplify \(\frac{9!}{6!}\)?
Answer
Expand only \(9!=9\cdot8\cdot7\cdot6!\), then cancel \(6!\).
Card 16 of 20
MediumStudy card 16
Prompt
What common error occurs when using a permutation for a committee?
Answer
The same unordered group is counted once for every possible ordering.
Card 17 of 20
HardStudy card 17
Prompt
How are \({}_nP_r\) and \({}_nC_r\) related?
Answer
\({}_nP_r=r!\,{}_nC_r\), because each unordered group has \(r!\) orders.
Formula
\[{}_nP_r=r!\,{}_nC_r\]
Card 18 of 20
EasyStudy card 18
Prompt
A code has 4 icon choices and 3 color choices. How many codes?
Answer
\(12\), by the Fundamental Counting Principle.
Card 19 of 20
MediumStudy card 19
Prompt
When should you avoid forcing \({}_nP_r\) or \({}_nC_r\)?
Answer
When the problem is naturally a sequence of independent stages whose legal choices can be multiplied directly.
Card 20 of 20
HardStudy card 20
Prompt
How do you solve a process that first chooses a team and then assigns roles?
Answer
Use a combination for the team, a permutation for the roles, then multiply the stages.
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