MATHCOUNTS Chapter 2026 Target Round (Problem 8)
The MATHCOUNTS Question Writing Committee is meeting to finalize the problems for competition year 2026. They are seated around a circular table where the seat of the committee chair, Greg, has already been determined in advance. The remaining six members must sit such that exactly one of the following pairs of writers must be seated next to each other: Thinula and Matthew, Ryoko and Michelle, or Rachel and Liam. In how many ways can the members be seated around the table?
Consider the 6 people to be Th, Ma, Ry, Mi, Ra, and Li. Since Greg's place is determined, we can consider placing the 6 members in a row.
: the number of permutations in which Th and Ma are adjacent.
: the number of permutations in which Ry and Mi are adjacent.
: the number of permutations in which Ra and Li are adjacent.
: the number of permutations in which Ry and Mi are adjacent.
: the number of permutations in which Ra and Li are adjacent.
By Hints 3, 4, 5, and 6,