A photo of four boys and four girls alternating

Question

Four distinct boys and four distinct girls stand in a row for a photo, arranged so that the genders strictly alternate (no two boys adjacent and no two girls adjacent).

How many such arrangements are there?

Accepted Answer

First decide the pattern of the row. With equal numbers, a strict alternation must be either B\,G\,B\,G\,B\,G\,B\,G \quad	ext{or}\quad G\,B\,G\,B\,G\,B\,G\,B, so there are 2 patterns. For a fixed pattern, the 4 boy-slots get the 4 distinct boys in 4! = 24 ways, and independently the 4 girl-slots get the 4 distinct girls in 4! = 24 ways. Multiply across the pattern choice too: 2 	imes 4! 	imes 4! = 2 	imes 24 	imes 24 = 1152. Takeaway. For alternating arrangements, count the valid patterns, then fill each group's slots independently: 2 \cdot 4! \cdot 4! = 1152.

A photo of four boys and four girls alternating

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