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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.
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Solution
In the 1st row, 6 teachers can be arranged among themselves in 6P6 i.e., 6! ways.
In the 2nd 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`
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