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 1^st row, 6 teachers can be arranged among themselves in ⁶P₆ i.e., 6! ways.

In the 2^nd row, 12 boys can be arranged among themselves in ¹²P₁₂ i.e., 12! ways.

So, there are 13 places group created by 12 boys in which 6 girls occupy any 6 places in ¹³P₆ ways.

∴ Required number of arrangements

= 6! × 12! × ¹³P₆

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

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

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

= `(12!13!)/7`