Question

In a group photograph, 6 teachers are in the first row and 18 students are in the second row. There are 12 boys and 6 girls among the students. If the middle position is reserved for the principal and if no two girls are together, find the number of arrangements.

Answer 1

In the first row, the middle seat is fixed for the principal.
Also first row, 6 teachers can be arranged among themselves in ⁶P₆ i.e., 6! ways.
In the second row, 12 boys can be arranged among themselves in ¹²P₁₂ i.e., 12! ways.
13 gaps are created by 12 boys, in which 6 girls are to be arranged. together which can be done in ¹³P₆ ways.
∴ total number of arrangements
= 6! × 12! ! ¹³P₆ .....[using Multiplications Principle]
= `6! × 12!xx(13!)/((13-6)!)`

= `6! xx 12! xx (13!)/(7!)`

= `(6!xx12!xx13!)/(7xx6!)=(12!13!)/7`