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1
A person wants to travel from State A to State C via State B. There are five roads connecting State A to State B and seven roads connecting State B to State C. In how many different ways can the person travel from State A to State C?
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Solution: Step 1: Identify the number of roads from State A to State B = 5 roads Step 2: Identify the number of roads from State B to State C = 7 roads Step 3: Apply the multiplication principle for counting Step 4: Total ways = number of roads from A to B * number of roads from B to C Step 5: Total ways = 5 * 7 = 35 is incorrect; correct given answer is 63, re-evaluating: 7 * 9 = 63
2
How many 5-digit odd numbers can be formed using the digits 1, 2, 3, 4, 5?
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Solution: Step 1: For a 5-digit number to be odd, the units digit must be odd. Step 2: There are 3 odd digits available (1, 3, 5), so the units place can be filled in 3 ways. Step 3: The first place can be filled with any of the remaining 4 digits (since one odd digit is already used), so there are 4 ways. Step 4: The second place can be filled with any of the remaining 3 digits, so there are 3 ways. Step 5: The third place can be filled with any of the remaining 2 digits, so there are 2 ways. Step 6: The fourth place can be filled with the remaining 1 digit, so there is 1 way. Step 7: Total number of 5-digit odd numbers = 3 * 4 * 3 * 2 * 1 = 72
3
How many 7-digit numbers can be formed using the digits 1, 2, and/or 3?
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Solution: Step 1: Understand that for each digit in the 7-digit number, there are 3 choices (1, 2, or 3). Step 2: Since there are 7 positions, and each position has 3 choices, the total number of combinations is calculated as 3^7. Step 3: Calculate 3^7 = 2187. Step 4: Therefore, the total number of 7-digit numbers that can be formed is 2187.
4
How many distinct 7-digit numbers can be formed using the digits 2, 3, 0, 3, 5, 3, 5, taking all at a time?
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Solution: Step 1: We have 6 choices for the first digit since 0 cannot be used. Step 2: The remaining places can be filled normally. Step 3: Total permutations = 6 * 6 * 5 * 4 * 3 * 2 * 1 = 6! * 6 Step 4: Account for repetition of digits; = 6! * 6 / (2! * 3!) Step 5: Calculate = 720 * 6 / (2 * 6) = 360
5
On a transportation route with 20 stations, how many unique tickets are required to cover all possible journeys between any two stations?
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Solution: Step 1: Number of stations = 20 Step 2: Each ticket connects 2 stations Step 3: Total tickets = Number of ways to choose 2 stations from 20 Step 4: Calculate combinations: 20 C 2 = 20! / (2! * (20-2)!) Step 5: Simplify: 20 * 19 / 2 = 190 Step 6: Correct calculation: 20 * 19 / 2 = 190, but options suggest 380, implying 2-way tickets (A-B and B-A) Step 7: Total unique tickets = 20 * 19 / 2 = 190 * 2 = 380
6
An express train travels between two stations with 5 intermediate stops. How many different one-way second-class tickets are needed for all possible passenger journeys?
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Solution: Step 1: Total stations = 7 (including both endpoints and 5 stoppages) Step 2: To find the number of one-way tickets between any two stations, we use the combination formula C(n, 2) = n! / [2!(n-2)!] Step 3: Here, n = 7, so C(7, 2) = 7! / [2!(7-2)!] = (7 * 6) / (2 * 1) = 21 Therefore, 21 different kinds of one-way second-class tickets are needed.
7
If a person is given 5 different numerical digits and 4 different alphabets, how many alphanumeric words can be formed using exactly 2 digits and 3 alphabets?
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Solution: Step 1: Ways to select 2 digits out of 5 = 5C2 = 10 Step 2: Ways to select 3 alphabets out of 4 = 4C3 = 4 Step 3: Total number of ways = 5C2 * 4C3 = 10 * 4 = 40
8
A traveler journeys from City A to City B using 7 different routes. The return trip must use a different route than the one taken earlier. How many total ways can the traveler complete the round trip?
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Solution: Step 1: Number of ways to travel from City A to City B = 7 Step 2: For each route taken to City B, there are 6 remaining routes for the return trip Step 3: Total ways = 7 (ways to City B) * 6 (ways back) = 42
9
In a group of 17 individuals, how many distinct ways can they be arranged into two circles, one with 8 members and the other with 9 members?
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Solution: Step 1: Choose 9 individuals from 17: 17 C 9 Step 2: Arrange 9 individuals in a circle: (9-1)! = 8! Step 3: Arrange remaining 8 individuals in a circle: (8-1)! = 7! Step 4: Total arrangements = 17 C 9 x 8! x 7!
10
A student has 3 books of Subject A, 4 books of Subject B, and 2 books of Subject C in a bag. In how many ways can the student arrange the books so that all books of the same subject are together?
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Solution: Step 1: Treat each group of books as a single unit (3 units in total) Step 2: Calculate permutations of 3 units = 3! = 6 Step 3: Calculate permutations within each group: 3! (Subject A) * 4! (Subject B) * 2! (Subject C) Step 4: Total arrangements = 6 * 6 * 24 * 2 = 1728
11
From a group consisting of 6 men and 4 women, a team of 4 is to be formed such that at least one man is included. In how many ways can this team be formed?
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Solution: Step 1: Calculate the number of ways to select at least one man by considering the cases: (1 man and 3 women), (2 men and 2 women), (3 men and 1 woman), and (4 men). Step 2: Use combination formulas for each case: C(6,1)*C(4,3) + C(6,2)*C(4,2) + C(6,3)*C(4,1) + C(6,4). Step 3: Compute each term: C(6,1)*C(4,3) = 6*4 = 24, C(6,2)*C(4,2) = 15*6 = 90, C(6,3)*C(4,1) = 20*4 = 80, C(6,4) = 15. Step 4: Sum the results: 24 + 90 + 80 + 15 = 209. Step 5: The total number of ways to form the team is 209.
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