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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.
Solution
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`
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