In how many different ways can five distinct persons stand in a single straight line for a group photograph?

Introduction / Context:
This problem is a direct application of the idea of permutations. We are asked to count the number of ways that five distinct people can be arranged in a straight line. Since each arrangement corresponds to a unique ordering of the five individuals, the question becomes a simple permutation of distinct objects taken all at once.

Given Data / Assumptions:

• There are 5 different persons.
• They are to stand in a single straight line.
• Every person is distinct so changing their positions creates a different arrangement.
• All 5 positions in the line must be filled.

Concept / Approach:
When arranging n distinct objects in a sequence, the total number of permutations is n factorial (written as n!). This is because there are n choices for the first position, n minus one choices for the second position, and so on down to 1 choice for the last position. For this problem, n equals 5. Hence, the result is 5! which counts all possible standing orders of the five people.

Step-by-Step Solution:
Step 1: Determine the number of options for the first position. There are 5 people, so there are 5 choices for the first spot. Step 2: Determine the options for the second position. After one person is placed, 4 people remain, so there are 4 choices for the second spot. Step 3: Continue this reasoning. Third spot: 3 choices; fourth spot: 2 choices; fifth spot: 1 choice. Step 4: Multiply the number of choices for each position. Total arrangements = 5 * 4 * 3 * 2 * 1 = 5! = 120.

Verification / Alternative check:
You can also express the count directly as the permutation formula P(5, 5) which equals 5!, because we are arranging all 5 people. Calculating 5! explicitly confirms 120 distinct arrangements. There is no constraint on who can stand where, so no further adjustments are needed.

Why Other Options Are Wrong:
The values 240, 360 and 720 are all multiples of 120 that would arise only if extra factors were mistakenly included, such as incorrectly counting additional positions. The value 60 is half of 120 and might come from stopping the multiplication too early or ignoring some positions. Only 120 corresponds exactly to 5!.

Common Pitfalls:
Some learners misinterpret the question and try to apply combinations, which count selections without order, rather than permutations, which count arrangements with order. Others may incorrectly think that rotating the whole line does not produce new arrangements, but in a line every distinct order counts separately. Remember that whenever order matters for distinct entities, factorial based permutations are usually the correct tool.

Final Answer:
Five distinct persons can stand in a straight line in 120 different ways.