Permutations & Combinations Question - None

Question

A condolence meeting is being held in a big hall which has 7 doors by which the mourners enter the hall. One can use any one of the 7 doors to enter and can come at anytime during the meeting. At each door a register is kept in which a mourner has to put down his signature while entering the hall. If 200 people attend the meeting, how many different sets of 7 lists of signatures can arise?

Accepted Answer

Step 1: Understand the problem structure

We have 200 distinct people and 7 distinct doors. Each person chooses exactly one door to enter. The signature lists at each door represent which people used that door, and the ORDER in each list matters (they sign sequentially).

Step 2: Model as function counting

This is equivalent to counting functions f: {200 people} → {7 doors}. Each person is assigned to exactly one door.

Step 3: Apply the multiplication principle

Person 1 can enter through any of 7 doors: 7 choices
Person 2 can enter through any of 7 doors: 7 choices
...
Person 200 can enter through any of 7 doors: 7 choices

Step 4: Calculate total arrangements

Total different sets of 7 signature lists = 7 × 7 × 7 × ... × 7 (200 times) = 7^200

∴ Answer: 7^200 (This is option B)

Note: The signature lists are distinguishable sets because the sequence and content of signatures at each door creates different outcomes. Each of the 200 people's choice of door independently contributes a factor of 7.

Question:

Step-by-Step Solution

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