Permutation: Level 1
1.
A person has 3 different shirts, 4 pairs of pants, and 2 pairs of shoes. How many different outfits can they create?
Solution (C): This is the Fundamental Counting Principle. We multiply the number of options for each choice.
Total outfits = (Number of shirts) \(\times\) (Number of pants) \(\times\) (Number of shoes)
Total outfits = \(3 \times 4 \times 2 = 24\).
2.
How many 3-digit security codes can be formed using the digits {1, 2, 3, 4} if repetition of digits is allowed?
Solution (B): There are 3 “slots” to fill.
Slot 1: 4 choices {1, 2, 3, 4}
Slot 2: 4 choices (repetition is allowed)
Slot 3: 4 choices (repetition is allowed)
Total = \(4 \times 4 \times 4 = 4^3 = 64\).
3.
How many 3-digit numbers can be formed using the digits {1, 2, 3, 4, 5} if no digit can be repeated?
Solution (A): This is a permutation without repetition.
Slot 1 (Hundreds): 5 choices {1, 2, 3, 4, 5}
Slot 2 (Tens): 4 choices left (since one digit is used)
Slot 3 (Units): 3 choices left
Total = \(5 \times 4 \times 3 = 60\).
4.
A quiz has 5 True/False questions. How many different ways are there to answer the entire quiz?
Solution (D): Each question has 2 possible answers (True or False).
Q1: 2 choices
Q2: 2 choices
Q3: 2 choices
Q4: 2 choices
Q5: 2 choices
Total ways = \(2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32\).
5.
In how many ways can 3 different books (A, B, C) be arranged on a shelf?
Solution (A): This is a permutation of 3 items.
Slot 1: 3 choices (A, B, or C)
Slot 2: 2 choices left
Slot 3: 1 choice left
Total = \(3 \times 2 \times 1 = 3! = 6\).
6.
How many 3-digit *positive* numbers can be formed using the digits {0, 1, 2, 3} if repetition of digits is allowed?
Solution (D): This problem has a constraint.
Slot 1 (Hundreds): 3 choices {1, 2, 3}. (Cannot be 0)
Slot 2 (Tens): 4 choices {0, 1, 2, 3}. (Repetition is allowed)
Slot 3 (Units): 4 choices {0, 1, 2, 3}.
Total = \(3 \times 4 \times 4 = 48\).
7.
A room has 5 doors. In how many ways can a person enter the room and leave the room, if they must use a different door to leave?
Solution (E):
Choice 1 (Enter): 5 options (any of the 5 doors).
Choice 2 (Leave): 4 options (any door *except* the one used to enter).
Total ways = \(5 \times 4 = 20\).
8.
How many 4-digit numbers can be formed using only the digits {8, 9} if repetition is allowed?
Solution (B): There are 4 slots to fill, and 2 choices for each.
Slot 1: 2 choices {8, 9}
Slot 2: 2 choices
Slot 3: 2 choices
Slot 4: 2 choices
Total = \(2 \times 2 \times 2 \times 2 = 2^4 = 16\).
9.
A person can travel from City A to City B by 3 routes, and from City B to City C by 4 routes. How many different routes can they take from City A to City C?
Solution (A): By the Fundamental Counting Principle, we multiply the number of choices for each leg of the journey.
Total routes = (Routes A to B) \(\times\) (Routes B to C)
Total routes = \(3 \times 4 = 12\).
10.
A test has 2 multiple-choice questions (each with 4 options) and 3 true/false questions (2 options). How many ways can the test be answered?
Solution (C): We find the total number of choices by multiplying the choices for each question.
MC 1: 4 choices
MC 2: 4 choices
T/F 1: 2 choices
T/F 2: 2 choices
T/F 3: 2 choices
Total = \(4 \times 4 \times 2 \times 2 \times 2 = 16 \times 8 = 128\).
Score: 0 / 10