Question

In how many ways $6$ students and $4$ teachers be arranged in a row so that no two teachers are together ?

• A. $604800$
• B. $246800$
• C. $258600$
• D. $55500$

Answer 1

The correct option is A $604800$

Here $4$ teachers position with $3$ students inserted between to keep teachers apart, These fase, only have to worry about placing the other $3$ students. We have $3$ students and $4$ teachers that can separated there, so these case $7$ position to deal with $3$ of these case students, So we get $7$ choose $3$.

Now we have deal with considering of specific teachers and students. The teachers can be ordered $4!$ ways and the students can be ordered $6!$ ways.

$\left(\begin{array}{c}7\\ 3\end{array}\right)$4!.6!$=\frac{7!}{3!.4!}\times 4!\times 6!$

$=\frac{7!\times 6!}{3!}$

$=\frac{7!\times 6\times 5\times 4\times 3!}{3!}$

$=7!\times 6\times 5\times 4$

$=6,04,800$