How many ways can the committee select the chairperson, secretary, and subcommittee?

Question

A committee of 28 people must select a chairperson and secretary. The committee must also form a subcommittee consisting of 5 people but the chairperson and secretary cannot serve on the committee

Answer 1

ways to pick 2 people from 28 = 28C2 = 28!/2!26! = 28(27)/2 = 14(27 = 378 ways

ways to pick 5 from 28 = 28C5 = 28!/5!23! = 28(27)(26)(25)(24)/5(4)(3)(2) = 28(27)(26)(5) = 98,280 ways to pick the 5

assuming order does not matter, otherwise calculate using permutations instead of combinations

but then there's that wrinkle about the 2 can't be in the 5

so reduce the ways a bit

maybe 28-2 = 26

then 26C5 = 26!/5!21! = 26(25)(24)(23)(22)/5(4)(3)2 = 26(5)23(22) = 65,780 ways to pick the 5

basic formula for Combinations is nCr = n!/r!(n-r)!