Question

1. A biking club consists of 25 people, 10 from California and 15 from Wisconsin. A
committee consisting of 5 Wisconsinites and 3 Californians is to be formed from the club
members.

a. How many different committees are possible if one of the Wisconsinites refuses to serve with
one of the Californians?

b. How many different committees are possible if two of the Wisconsinites will only serve if
they are together?

Answer 1

1)a)

number of total committees possible without any constraint =(₁₀C₃)*(₁₅C₅) =120*3003 =360360

number of committees so that that particular Wisconsinites and Californians are in the committee

=(₉C₂)*(₁₄C₄) =120*3003 =36*1001 =36036

therefore possible committee so that one of the Wisconsinites refuses to serve with
one of the Californians =360360-36036 =324324

b)

here number of ways =N(both of them are selected)+N(none of them are selected)

=(₁₀C₃)*(₁₃C₃)+(₁₀C₃)*(₁₃C₅) =120*286+120*1287 =188760