Question

25 distinct objects are divided into 5 groups and each group consists of exactly 5 objects. Then the number of ways of forming such groups, is

• A. $\frac{25!}{(5!)^5}$
• B. $\frac{25!}{5!}$
• C. $\frac{25!}{(5!)^6}$
• D. $\frac{25!}{(5!)^4}$
• E. $\frac{25!}{(5!)^3}$

Hint:

Always divide by $n!$ when the groups are identical (unnamed) to avoid overcounting permutations of the groups.

Answer 1

The Correct Option is C

Step 1: Concept
The number of ways to divide $mn$ distinct objects into $n$ equal groups of size $m$ is $\frac{(mn)!}{(m!)^n \cdot n!}$.

Step 2: Analysis
Here, total objects $mn = 25$, group size $m = 5$, and number of groups $n = 5$.

Step 3: Calculation
Number of ways $= \frac{25!}{(5!)^5 \cdot 5!} = \frac{25!}{(5!)^6}$. Final Answer: (C)