Neha arranges 10 distinct numbers with 10 distinct functions, so that each…

Concept:

The Fundamental Counting Principle governs assignments: when n distinct items are placed into n distinct positions one item per position (a one-to-one assignment, i.e. a bijection), the total number of ways is n factorial (n!) = n × (n − 1) × (n − 2) × ... × 1. This is a different count from the total number of functions from an n-element set to an n-element set, which allows repeats and equals n raised to the power n.

Application:

Here Neha places each of her 10 distinct numbers into one of the 10 distinct functions, one number per function — a one-to-one assignment, since "arranges" means every number is placed in exactly one position and every position gets exactly one number.

1. Number of choices for the 1st function: 10 (any of the 10 numbers).

2. Number of choices for the 2nd function: 9 (one number is already placed).

3. Number of choices for the 3rd function: 8, and so on, decreasing by one each time.

4. Continue down to the 10th function, which has only 1 remaining number left to place.

5. Multiply all ten choices together: 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 10!.

6. Evaluate: 10! = 3,628,800.

Cross-check:

Cross-check by pairing factors: (10 × 9) × (8 × 7) × (6 × 5) × (4 × 3) × (2 × 1) = 90 × 56 × 30 × 12 × 2 = 3,628,800, the same result. Contrast this one-to-one model with the option written as 10 raised to the power 10: that value is the count of assignments where each function may independently pick any of the 10 numbers with repetition allowed, which is a different scenario from arranging 10 distinct numbers one-to-one.

Since 3,628,800 does not equal any of the specific numeric values offered, the correct total falls under "None of these."