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Question: The total number of ways of dividing mn objects into n equal groups, when the groups are distinguish...

The total number of ways of dividing mn objects into n equal groups, when the groups are distinguishable is

A

∠mn‾(∠m)n∠n\frac{\angle\underline{mn}}{(\angle m)^{n}\angle n}

B

∠mn‾(∠m)nn\frac{\angle\underline{mn}}{(\angle m)^{n}n}

C

∠mn‾ 6mu∠n(∠m)n\frac{\angle\underline{mn}\mspace{6mu}\angle n}{(\angle m)^{n}}

D

∠mn‾(∠m)n\frac{\angle\underline{mn}}{(\angle m)^{n}}

Answer

∠mn‾(∠m)n\frac{\angle\underline{mn}}{(\angle m)^{n}}

Explanation

Solution

In this case the groups are distinguishable, therefore, the required number

= mnCmxmn−mCmxmn−2mCmx...x2mCmxmCmmnC_{m}x^{mn - m}C_{m}x^{mn - 2m}C_{m}x...x^{2m}C_{m}x^{m}C_{m}

(Note that each group contains m objects)

= ∠mn(∠m)n\frac{\angle mn}{(\angle m)^{n}}