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Question #40643

Difficulty: 8/10 | 🟢 Free

The Question

A university has 25 new students who need to be assigned to 5 distinct dormitories, denoted as $D_1, D_2, D_3, D_4, D_5$ , with exactly 5 students in each dormitory. Among the 25 students, there are 5 specific students $S_1, S_2, S_3, S_4, S_5$ . Let $N$ be the total number of ways to assign the students such that for all $i \in \{1, 2, 3, 4, 5\}$ , student $S_i$ is NOT assigned to dormitory $D_i$ . Find the exponent of the highest power of 5 that divides $N$ .

Options

A.
1
B.
2
C.
3
D.
4

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Multinomial Theorem|Exponent of Prime in n!|Derangements (Inclusion-Exclusion)

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of the Multinomial Theorem, Inclusion-Exclusion (Derangements), and Legendre's Formula.

The primary trap lies in the assumption that derangement probabilities can be applied independently of the multinomial group sizes, leading students to incorrectly multiply the total ways by a standard derangement fraction.

Students must rigorously set up the Inclusion-Exclusion principle for multisets.

The architectural elegance here is the exact cancellation of the $j=0$ and $j=1$ terms, which rewards students who algebraically factor the expression rather than attempting brute-force estimation.

Finally, applying Legendre's formula to a non-integer factorial fraction combined with the denominator $5^2$ from the inclusion-exclusion sum isolates the true exponent of the prime.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Multinomial Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free