**Question:**

when a group photograph is taken, all the seven teachers should be in the first row, and all the twenty students should be in the second row. If the tow corners of the second row are reserved for the two tallest students, interchangeable only between them, and if the middle seat of the front row is reserved for the principal, how many arrangements are possible?

**Solution:**

For the first row:

There are 7 teachers in which the position of principal is fixed.

Therefore, the teachers can be arranged in $\mathrm{p}(7,7)=5040$.

For the second row:

The tallest students are at the ends and can be arranged in $2 !=2$ ways.

Rest 18 students can be arranged in $\mathrm{P}(18,18)$ ways.

Formula:

Number of permutations of $n$ distinct objects among $r$ different places, where repetition is not allowed, is

$P(n, r)=n ! /(n-r) !$

Therefore, permutation of 18 different objects in 18 places is

$P(18,18)=\frac{18 !}{(18-18) !}$

$=\frac{18 !}{0 !}=\frac{18 !}{1}=18 !$

Therefore, a total number of arrangements of the second row is $2 \times 18 !$

Total arrangements $=2 \times 18 ! \times 5040=10080 \times 18 !$

The total number of arrangements is $10080 \times 18 !$