The annual National No Spying Day is celebrated at KAOS headquarters this year. There are 11 Control agents and 17 KAOS agents attending. How many ways can we choose a team of 5 agents if 3 team members need to be from Control and 2 from KAOS? How many ways can we choose a team of 7 agents if at least 1 team member needs to be from Control?

Answer:a) There are 22440 ways of choosing a team if 3 team members need to be from Control and 2 from KAOS.b) There are 11,820,992 ways of choosing a team if at least 1 team member needs to be from Control.Step-by-step explanation:The group is formed by 11 control agents and 17 KAOS agents. Since the order in which we select the members of the group doesn't really matter, we're going to use Combinations. Note that the Combinations formula for [tex]C_{n|r}[/tex] is [tex]\frac{n!}{r!(n-r)!}[/tex]a) The way of choosing  a team of agents if 3 team members need to be from Control and 2 from KAOS is:[tex]C_{11|3}[/tex] × [tex]C_{17|2}[/tex] = [tex]\frac{11!}{3!8!}[/tex] × [tex]\frac{17!}{2!15!}[/tex] = 165 × 136 = 22440b) The way of choosing a team of 7 agents if at least 1 team member needs to be from Control. To do this calculation easier, we can rewrite the question as "calculating how many teams of 7 agents we can make in total and then subtracting the number of teams with no members from Control"This could be written as [tex]C_{28|7}-C_{17|7} = [tex]\frac{28!}{7!21!}[/tex]- [tex]\frac{17!}{7!10!}[/tex] = 1184040-19448 = 11820992[/tex]