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A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.
Q. What is the minimum possible number of different types of prizes?
Correct answer is '2'. Can you explain this answer?
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A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a r...
It is given that the most expensive item is a diamond ring of type a and there is exactly one of these. Since the item b should be at least twice. The minimum number of items will be obtained when a=1 and b=99, which means there are only two different types of items.
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A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a r...
The Minimum Possible Number of Different Types of Prizes is 2
To determine the minimum possible number of different types of prizes, let's consider the scenario where there are more than two types of prizes.
Assuming Three Types of Prizes:
If we assume that there are three types of prizes, a, b, and c, with a being the most valuable, we need to consider the minimum number of items in each type.
- Type a: 1 item (diamond ring)
- Type b: At least 2 items (since it is stated that the number of items doubles as we move to the next type)
- Type c: At least 4 items (twice the number of type b items)
Now, let's calculate the total number of items in these three types:
Total items = 1 + 2 + 4 = 7
However, there are only 100 boxes available, and the total number of items in three types is only 7. This implies that there are many empty boxes, which contradicts the given information that each box contains a mystery prize. Therefore, it is not possible to have three types of prizes.
Assuming Two Types of Prizes:
Now, let's consider the scenario where there are only two types of prizes, a and b.
- Type a: 1 item (diamond ring)
- Type b: At least 2 items (twice the number of type a items)
Calculating the total number of items in these two types:
Total items = 1 + 2 = 3
With only two types of prizes, the total number of items is 3, which is less than the total number of boxes (100). This means that it is possible to distribute the items in the boxes without any empty boxes.
Therefore, the minimum possible number of different types of prizes is 2.
Key Points:
- Assuming three types of prizes results in empty boxes, contradicting the given information.
- Assuming two types of prizes allows for the distribution of all items without any empty boxes.
- Therefore, the minimum possible number of different types of prizes is 2.
To determine the minimum possible number of different types of prizes, let's consider the scenario where there are more than two types of prizes.
Assuming Three Types of Prizes:
If we assume that there are three types of prizes, a, b, and c, with a being the most valuable, we need to consider the minimum number of items in each type.
- Type a: 1 item (diamond ring)
- Type b: At least 2 items (since it is stated that the number of items doubles as we move to the next type)
- Type c: At least 4 items (twice the number of type b items)
Now, let's calculate the total number of items in these three types:
Total items = 1 + 2 + 4 = 7
However, there are only 100 boxes available, and the total number of items in three types is only 7. This implies that there are many empty boxes, which contradicts the given information that each box contains a mystery prize. Therefore, it is not possible to have three types of prizes.
Assuming Two Types of Prizes:
Now, let's consider the scenario where there are only two types of prizes, a and b.
- Type a: 1 item (diamond ring)
- Type b: At least 2 items (twice the number of type a items)
Calculating the total number of items in these two types:
Total items = 1 + 2 = 3
With only two types of prizes, the total number of items is 3, which is less than the total number of boxes (100). This means that it is possible to distribute the items in the boxes without any empty boxes.
Therefore, the minimum possible number of different types of prizes is 2.
Key Points:
- Assuming three types of prizes results in empty boxes, contradicting the given information.
- Assuming two types of prizes allows for the distribution of all items without any empty boxes.
- Therefore, the minimum possible number of different types of prizes is 2.
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A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.Q. What is the minimum possible number of different types of prizes?Correct answer is '2'. Can you explain this answer?
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A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.Q. What is the minimum possible number of different types of prizes?Correct answer is '2'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.Q. What is the minimum possible number of different types of prizes?Correct answer is '2'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.Q. What is the minimum possible number of different types of prizes?Correct answer is '2'. Can you explain this answer?.
A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.Q. What is the minimum possible number of different types of prizes?Correct answer is '2'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.Q. What is the minimum possible number of different types of prizes?Correct answer is '2'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.Q. What is the minimum possible number of different types of prizes?Correct answer is '2'. Can you explain this answer?.
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