Question

A Senate committee has 4 Democrats and 3 Republicans. A secretary, treasurer, and sergeant-at-arms must be named from the committee members in such a way that each party gets at least one of the three offices. If each office must be occupied by a distinct member, how many ways are there to name people to the three offices?

(A) 204 (B) 30 (C) 6 (D) 24 (E) 180 (F) 210 (G) 72 (H) 186

This is on a practice exam for my finite class and I have no clue how to solve this

Answer 1

number of ways =N(select 2 person from Democrat and 1 person from Republicans and assign them roles)+N(select 1 person from Democrat and 2 person from Republicans and assign them roles)

=(₄C₂)*(₃C₁)*3!+(₄C₁)*(₃C₂)*3!=6*3*6+4*3*6 =180

option E is correct