Question

The number of ways of dividing \(200\) dissimilar things into \(10\) groups each containing \(20\) elements is:

• A. \[ \frac{200!}{(20!)^{10}\,10!} \]
• B. \[ \frac{200!}{(10!)^{10}\,20!} \]
• C. \[ \frac{200!}{(20!)^{10}\,10!} \]
• D. \[ \frac{200!}{(10!)^{20}\,20!} \]

Hint:

When dividing distinct objects into equal-sized unlabelled groups, first divide by the factorials of the group sizes and then divide by the factorial of the number of groups.

Answer 1

The Correct Option is C

Step 1: Partition 200 distinct objects into 10 ordered groups of 20.
The number of ways to sequentially assign objects to 10 labelled groups of equal size is 200!/(20!)¹⁰, where each (20!) removes internal ordering within a group.

Step 2: Account for indistinguishable groups.
Since the 10 groups are unlabelled, any permutation of the groups represents the same division. There are 10! such permutations.

Step 3: Write the final expression.
Dividing by 10! corrects for the overcounting, yielding 200!/[(20!)¹⁰·10!].

Step 4: Final conclusion.
The required number of ways is 200!/[(20!)¹⁰·10!].