Question

A reserve of 12 railway station masters is to be divided into two groups of 6 each one for day duty and the other for night duty. The number of ways in which this can be done if two specified persons A, B should not be included in the same group is

• A. 500
• B. 504
• C. 508
• D. 512

Answer 1

The correct option is B 504

If $A$ is in the day duty party, then $B$ must be in night duty party.
To complete the party, we must introduce $5$ more to the first party.
These $5$ can be chosen from the remaining $10$

This can be done in ${}^{10}{C}_5$ ways
This automatically leaves the remaining $5$ to the night duty.
So, with $A$ in the night duty the party can be formed in ${}^{10}{C}_5$ ways.
Suppose then $A$ is in the night duty.

This is equivalent to the whole batch being interchanged.
So we can have ${}^{10}{C}_5$ more ways of forming the batches.
The total number required is, therefore, equal to ${}^{10}{C}_5+{}^{10}{C}_5+{}^{10}{C}_5=2.\frac{\mathrm{10.9.8.7.6}}{\mathrm{1.2.3.4.5}}=\mathrm{9.8.7}=504$