1) 우산문제 답변

To find the probability that only two people took their correct umbrellas when all three umbrellas are picked randomly, we'll analyze the possible arrangements of umbrellas.

### Problem Breakdown
1. **Total Umbrellas:** 3
2. **Owners:** A, B, and C
3. **Correct Umbrella Assignment:**
   - A owns umbrella 1
   - B owns umbrella 2
   - C owns umbrella 3

### Derangement Calculation
A derangement is a permutation where none of the items appear in their original positions. In this problem, we are interested in finding permutations where exactly two of the three owners end up with their own umbrellas.

Let's denote the umbrellas as 1 (A's umbrella), 2 (B's umbrella), and 3 (C's umbrella).

### Possible Cases
1. **Case 1:** A and B get the correct umbrella, but C does not.
   - A gets umbrella 1.
   - B gets umbrella 2.
   - Since umbrella 3 is the only one left, C must get umbrella 3, which is C's correct umbrella.
   - This case is not possible because all three people will have their correct umbrellas.

2. **Case 2:** A and C get the correct umbrella, but B does not.
   - A gets umbrella 1.
   - C gets umbrella 3.
   - Since umbrella 2 is the only one left, B must get umbrella 2, which is B's correct umbrella.
   - This case is not possible because all three people will have their correct umbrellas.

3. **Case 3:** B and C get the correct umbrella, but A does not.
   - B gets umbrella 2.
   - C gets umbrella 3.
   - Since umbrella 1 is the only one left, A must get umbrella 1, which is A's correct umbrella.
   - This case is not possible because all three people will have their correct umbrellas.

4. **Case 4:** Only A gets the correct umbrella.
   - A gets umbrella 1.
   - B and C must swap their umbrellas.
   - B gets umbrella 3 (C's umbrella).
   - C gets umbrella 2 (B's umbrella).

5. **Case 5:** Only B gets the correct umbrella.
   - B gets umbrella 2.
   - A and C must swap their umbrellas.
   - A gets umbrella 3 (C's umbrella).
   - C gets umbrella 1 (A's umbrella).

6. **Case 6:** Only C gets the correct umbrella.
   - C gets umbrella 3.
   - A and B must swap their umbrellas.
   - A gets umbrella 2 (B's umbrella).
   - B gets umbrella 1 (A's umbrella).

### Total Number of Permutations
- The total number of ways to assign 3 umbrellas is ( 3! = 6 ).

### Favorable Outcomes
- Case 4: Only A gets the correct umbrella.
- Case 5: Only B gets the correct umbrella.
- Case 6: Only C gets the correct umbrella.

Each of these cases represents one favorable outcome.

### Probability Calculation
The probability that only two people took their correct umbrellas is:
[
frac{3}{6} = frac{1}{2}
]

-------------------


2) 속도문제 답변


Let's analyze this problem step by step to find the solution.

### Problem Breakdown
1. **Definitions:**
   - Let ( v_1 = 100 , mathrm{km/h} ) be the speed from A to B.
   - Let ( v_2 ) be the return speed from B to A.
   - The distance between A and B is ( d ) km.
   - The time taken to travel from A to B is ( t_1 ) hours.
   - The time taken to travel from B to A is ( t_2 ) hours.

2. **Formulas:**
   - Time taken from A to B:
   [
   t_1 = frac{d}{v_1}
   ]
   - Time taken from B to A:
   [
   t_2 = frac{d}{v_2}
   ]
   - Total time for the round trip:
   [
   t_{mathrm{total}} = t_1 + t_2 = frac{d}{v_1} + frac{d}{v_2}
   ]
   - Total distance for the round trip:
   [
   d_{mathrm{total}} = 2d
   ]
   - Average speed for the round trip:
   [
   v_{mathrm{avg}} = frac{d_{mathrm{total}}}{t_{mathrm{total}}} = frac{2d}{frac{d}{v_1} + frac{d}{v_2}} = frac{2v_1v_2}{v_1 + v_2}
   ]

### Problem Statement
We want to find the return speed ( v_2 ) such that the average speed for the entire round trip is 200 km/h.

### Solution Steps
1. Substitute ( v_1 = 100 , mathrm{km/h} ) and ( v_{mathrm{avg}} = 200 , mathrm{km/h} ) into the average speed formula:
   [
   frac{2 cdot 100 cdot v_2}{100 + v_2} = 200
   ]

2. Simplify the equation:
   [
   200v_2 = 20000 + 2 cdot 100 cdot v_2
   ]
   [
   200v_2 = 20000 + 200v_2
   ]

3. Isolate ( v_2 ):
   [
   0 = 20000
   ]

### Conclusion
The equation simplifies to an impossible statement because it requires infinite speed. Therefore, it is impossible to achieve an average round-trip speed of 200 km/h if the speed from A to B is 100 km/h.