Question

A team 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 $A$ and $B$ should not be included in the same group is

• A. $420$
• B. $504$
• C. $714$
• D. $924$

Answer 1

The correct option is B $504$

Total ways without restriction $={}^{12}{C}_6\times {}^6{C}_6$
Total ways when $A$ and $B$ are together $=2\times {}^{10}{C}_4\times {}^6{C}_6$
$\therefore$ Total required ways
$={}^{12}{C}_6\times {}^6{C}_6-2\times {}^{10}{C}_4\times {}^6{C}_6=\frac{1}{6!}[\frac{12\times 11\times 10!}{6\times 5\times 4!}-\frac{2\times 10!}{4!}]=\frac{1}{6!}\left[\frac{12\times 10!}{5\times 4!}\right]=9\times 8\times 7=504$