Arrangements With Together Condition

When a subset of objects must stay together, they are treated as one combined block, then permuted internally, a reliable JEE Advanced grouping method. Tie the 3 girls into a single block, so there are now 5 boys plus 1 girl-block, totalling 6 units to arrange in a row in 6! = 720 ways. Within the block, the 3 girls can be arranged among themselves in 3! = 6 ways. By the multiplication principle, the total is 720 × 6 = 4320. Option 720 counts only the block arrangements without internal girl orderings. Option 2880 uses an incorrect internal factor. Option 40320 = 8! ignores the togetherness condition entirely. Hence there are 4320 arrangements. Plausibility check: the count must be smaller than the unrestricted 8! = 40320, and 4320 is exactly 8!/ (something), specifically the fraction where the girls are bound, consistent with treating three of eight items as a single movable unit. The handshake count is the prototypical example of a triangular number arising from pair selection, and inverting it produces a quadratic whose positive root gives the group size. This same C(n,2) structure underlies counting edges of a complete graph, diagonals of polygons, and chords of a circle, making it one of the most transferable identities in combinatorics.