Four friends A, B, C and D need to cross a bridge. A maximum of two persons can cross it at a time. It is night and they just have one lamp. Persons that cross the bridge must carry the lamp to find the way. A pair must walk together at the speed of slower person, After crossing the bridge, the person having faster speed in the pair will return with the lamp each time to accompany another person in the group. Finally, the lamp has to be returned at the original place and the person who returns the lamp has to cross the bridge again without lamp. To cross the bridge, the time taken by them is as follows: A : 1 minute, B : 2 minutes, C : 7 minutes and D : 10 minutes. What is the total minimum time required by all the friends to cross the bridge ?

The correct option is 23 minutes.

Step-by-Step Explanation:

To find the minimum time required for all four friends to cross the bridge, we need to carefully follow the rules given in the problem:
1. A maximum of two people can cross the bridge at a time.
2. They must carry the single available lamp to cross.
3. A pair crossing together walks at the speed of the slower person.
4. After crossing, the faster person in the pair must return with the lamp to the starting side.
5. At the end, the lamp must be returned to the original side (start), and the person who returns it must cross the bridge once more without the lamp.

Let the times taken by the friends be:
A = 1 minute (Fastest)
B = 2 minutes
C = 7 minutes
D = 10 minutes (Slowest)

Let's map out the optimal sequence of moves to minimize the crossing time:

Step 1: A and B cross the bridge together.
Since they walk at the speed of the slower person (B), the time taken is:
Time = 2 minutes.
A and B are now on the other side.

Step 2: The faster person (A) returns with the lamp.
Time = 1 minute.
A is back at the start; B is on the other side.

Step 3: A and C cross the bridge together.
Since they walk at the speed of the slower person (C), the time taken is:
Time = 7 minutes.
A, B, and C are now on the other side.

Step 4: The faster person (A) returns with the lamp.
Time = 1 minute.
A is back at the start; B and C are on the other side.

Step 5: A and D cross the bridge together.
Since they walk at the speed of the slower person (D), the time taken is:
Time = 10 minutes.
All four (A, B, C, D) are now on the other side.

Step 6: Returning the lamp to the original place.
The lamp must be returned to the start. The fastest person (A) returns the lamp:
Time = 1 minute.
A is at the start with the lamp; B, C, and D are on the other side.

Step 7: The person who returned the lamp (A) crosses the bridge again without the lamp.
A crosses the bridge to join the others:
Time = 1 minute.

Total Time Calculation:

We sum the times taken for each step:

$$\text{Total Time}=2+1+7+1+10+1+1=23\text{ minutes}$$

Thus, the minimum total time required for all the friends to cross the bridge is 23 minutes.