Practice
Permutations and Combinations Practice
Fifty original questions on stage counting, factorials, permutations, combinations, method selection, and mixed counting.
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Question 1
Explanation
The choices occur in sequential stages, so multiply: \(3\times5=15\).
Question 2
Explanation
The choices occur in sequential stages, so multiply: \(4\times3\times2=24\).
Question 3
Explanation
The choices occur in sequential stages, so multiply: \(6\times2=12\).
Question 4
Explanation
The choices occur in sequential stages, so multiply: \(5\times4\times3=60\).
Question 5
Explanation
The choices occur in sequential stages, so multiply: \(2\times7\times2=28\).
Question 6
Explanation
Expand the factorial through \(1\): \(4!=4\cdot3\cdot2\cdot1=24\).
Question 7
Explanation
Expand the factorial through \(1\): \(5!=5\cdot4\cdot3\cdot2\cdot1=120\).
Question 8
Explanation
Expand the factorial through \(1\): \(6!=6\cdot5\cdot4\cdot3\cdot2\cdot1=720\).
Question 9
Explanation
Expand the factorial through \(1\): \(7!=7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=5040\).
Question 10
Explanation
Expand the factorial through \(1\): \(8!=8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=40320\).
Question 11
Explanation
Combination is appropriate because only the selected membership matters.
Question 12
Explanation
Permutation is appropriate because changing the order or position creates a different result.
Question 13
Explanation
Combination is appropriate because only the selected membership matters.
Question 14
Explanation
Permutation is appropriate because changing the order or position creates a different result.
Question 15
Explanation
Combination is appropriate because only the selected membership matters.
Question 16
Explanation
The ordered slots have \(6, 5, 4\) choices, so the total is \(6\times5\times4=120\).
Question 17
Explanation
The ordered slots have \(7, 6\) choices, so the total is \(7\times6=42\).
Question 18
Explanation
The ordered slots have \(8, 7, 6, 5\) choices, so the total is \(8\times7\times6\times5=1680\).
Question 19
Explanation
The ordered slots have \(5, 4, 3\) choices, so the total is \(5\times4\times3=60\).
Question 20
Explanation
The ordered slots have \(9, 8\) choices, so the total is \(9\times8=72\).
Question 21
Explanation
Ranked positions make order matter, so use \({}_{7}P_{3}=\frac{7!}{(7-3)!}=210\).
Question 22
Explanation
Ranked positions make order matter, so use \({}_{9}P_{2}=\frac{9!}{(9-2)!}=72\).
Question 23
Explanation
Ranked positions make order matter, so use \({}_{8}P_{5}=\frac{8!}{(8-5)!}=6720\).
Question 24
Explanation
Ranked positions make order matter, so use \({}_{10}P_{4}=\frac{10!}{(10-4)!}=5040\).
Question 25
Explanation
Ranked positions make order matter, so use \({}_{6}P_{4}=\frac{6!}{(6-4)!}=360\).
Question 26
Explanation
Only membership matters, so use \({}_{7}C_{2}=\frac{7!}{2!(7-2)!}=21\).
Question 27
Explanation
Only membership matters, so use \({}_{8}C_{3}=\frac{8!}{3!(8-3)!}=56\).
Question 28
Explanation
Only membership matters, so use \({}_{9}C_{4}=\frac{9!}{4!(9-4)!}=126\).
Question 29
Explanation
Only membership matters, so use \({}_{10}C_{3}=\frac{10!}{3!(10-3)!}=120\).
Question 30
Explanation
Only membership matters, so use \({}_{11}C_{2}=\frac{11!}{2!(11-2)!}=55\).
Question 31
Explanation
First choose the unordered group in \({}_{8}C_{3}\) ways. Then assign the role in \({}_{3}P_{1}\) ways. Multiplying gives \(168\).
Question 32
Explanation
First choose the unordered group in \({}_{9}C_{4}\) ways. Then assign the role in \({}_{4}P_{1}\) ways. Multiplying gives \(504\).
Question 33
Explanation
First choose the unordered group in \({}_{7}C_{3}\) ways. Then assign the roles in \({}_{3}P_{2}\) ways. Multiplying gives \(210\).
Question 34
Explanation
First choose the unordered group in \({}_{10}C_{2}\) ways. Then assign the role in \({}_{2}P_{1}\) ways. Multiplying gives \(90\).
Question 35
Explanation
First choose the unordered group in \({}_{11}C_{3}\) ways. Then assign the role in \({}_{3}P_{1}\) ways. Multiplying gives \(495\).
Question 36
Explanation
Translate the stated total into a multiplication, permutation, or combination equation and test the possible whole-number value.
Question 37
Explanation
Translate the stated total into a multiplication, permutation, or combination equation and test the possible whole-number value.
Question 38
Explanation
Translate the stated total into a multiplication, permutation, or combination equation and test the possible whole-number value.
Question 39
Explanation
Translate the stated total into a multiplication, permutation, or combination equation and test the possible whole-number value.
Question 40
Explanation
Every unordered pair has two possible orders, so \({}_8P_2=2!\,{}_8C_2\).
Question 41
Explanation
Choose the group, assign the ordered roles, and apply the final stage: \({}_{10}C_{4}\cdot{}_{4}P_{2}\cdot1=2520\).
Question 42
Explanation
Choose the group, assign the ordered roles, and apply the final stage: \({}_{9}C_{3}\cdot{}_{3}P_{1}\cdot2=504\).
Question 43
Explanation
Choose the group, assign the ordered roles, and apply the final stage: \({}_{8}C_{2}\cdot{}_{2}P_{2}\cdot3=168\).
Question 44
Explanation
Choose the group, assign the ordered roles, and apply the final stage: \({}_{11}C_{5}\cdot{}_{5}P_{1}\cdot1=2310\).
Question 45
Explanation
Choose the group, assign the ordered roles, and apply the final stage: \({}_{7}C_{3}\cdot{}_{3}P_{2}\cdot2=420\).
Question 46
Explanation
The outcome is ordered, so the correct count is \(504\). The repetition condition is applied exactly as stated.
Question 47
Explanation
The outcome is an unordered group, so the correct count is \(70\). The repetition condition is applied exactly as stated.
Question 48
Explanation
The outcome is ordered, so the correct count is \(120\). The repetition condition is applied exactly as stated.
Question 49
Explanation
The outcome is ordered, so the correct count is \(450\). The repetition condition is applied exactly as stated.
Question 50
Explanation
The outcome is ordered, so the correct count is \(1320\). The repetition condition is applied exactly as stated.
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Questions to review
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- Question 1Fundamental Counting PrincipleEasy
- Question 2Fundamental Counting PrincipleEasy
- Question 3Fundamental Counting PrincipleEasy
- Question 4Fundamental Counting PrincipleEasy
- Question 5Fundamental Counting PrincipleEasy
- Question 6Factorial evaluationEasy
- Question 7Factorial evaluationEasy
- Question 8Factorial evaluationEasy
- Question 9Factorial evaluationEasy
- Question 10Factorial evaluationEasy
- Question 11Choose permutation or combinationEasy
- Question 12Choose permutation or combinationEasy
- Question 13Choose permutation or combinationEasy
- Question 14Choose permutation or combinationEasy
- Question 15Choose permutation or combinationEasy
- Question 16Sequential choices without repetitionMedium
- Question 17Sequential choices without repetitionMedium
- Question 18Sequential choices without repetitionMedium
- Question 19Sequential choices without repetitionMedium
- Question 20Sequential choices without repetitionMedium
- Question 21Permutation calculationMedium
- Question 22Permutation calculationMedium
- Question 23Permutation calculationMedium
- Question 24Permutation calculationMedium
- Question 25Permutation calculationMedium
- Question 26Combination calculationMedium
- Question 27Combination calculationMedium
- Question 28Combination calculationMedium
- Question 29Combination calculationMedium
- Question 30Combination calculationMedium
- Question 31Mixed combination and permutationMedium
- Question 32Mixed combination and permutationMedium
- Question 33Mixed combination and permutationMedium
- Question 34Mixed combination and permutationMedium
- Question 35Mixed combination and permutationMedium
- Question 36Reverse counting problemsMedium
- Question 37Reverse counting problemsMedium
- Question 38Reverse counting problemsMedium
- Question 39Reverse counting problemsMedium
- Question 40Reverse counting problemsMedium
- Question 41Multi-step countingHard
- Question 42Multi-step countingHard
- Question 43Multi-step countingHard
- Question 44Multi-step countingHard
- Question 45Multi-step countingHard
- Question 46Counting error analysisHard
- Question 47Counting error analysisHard
- Question 48Counting error analysisHard
- Question 49Counting error analysisHard
- Question 50Counting error analysisHard