The number of possible singing orders for 7 distinct players when the youngest player is not allowed to sing last

Introduction / Context:
This question tests the concept of permutations with an additional restriction on the position of one particular player. Such questions are common in aptitude and placement exams because they combine basic factorial ideas with logical counting of restricted arrangements.

Given Data / Assumptions:

• There are 7 distinct players who will sing one after another.
• One of these players is the youngest player.
• The youngest player must not be in the last position.
• All arrangements are simple linear orders with no ties and no repeats.

Concept / Approach:
Without restrictions, the number of ways to arrange n distinct people in a row is n factorial. When there is a restriction on the position of one person, we normally count total arrangements and subtract the arrangements that violate the restriction. This is called the complement method.

Step-by-Step Solution:
Step 1: Total unrestricted arrangements of 7 distinct players = 7!.Step 2: Compute 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040.Step 3: Count the arrangements where the youngest player is fixed in the last position.Step 4: If the youngest is last, the remaining 6 players can be arranged in the first 6 positions in 6! ways.Step 5: Compute 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720.Step 6: The number of valid arrangements is total arrangements minus invalid arrangements.Step 7: Valid arrangements = 7! - 6! = 5040 - 720 = 4320.

Verification / Alternative check:
You can also think of this as placing the youngest in any of the first 6 positions. First choose a position for the youngest in 6 ways, then arrange the remaining 6 players in the remaining 6 positions in 6! ways. So total valid arrangements = 6 * 6! = 6 * 720 = 4320, which matches the previous calculation.

Why Other Options Are Wrong:

• 2580 is much smaller than 4320 and does not correspond to any standard factorial based count for 7 people.
• 3687 is not close to any natural expression like 7!, 7! - 6! or 6 * 6! and seems arbitrary.
• 5460 is larger than the total number of possible permutations 7!, so it is impossible.

Common Pitfalls:
Candidates often simply calculate 7! and stop, forgetting the restriction on the youngest player. Another common error is to try to directly count arrangements where the youngest is in allowed positions but then forget to multiply by all permutations of the remaining players. The safest method is to use the complement approach: count all arrangements and subtract the ones that violate the condition.

Final Answer:
The youngest player can sing in 4320 different valid orders, so the correct answer is 4320.