16 people ( 10 men & 6 women) are on a team. How many different
ways can the team chose a captain, manager, assistant coach if:
Rule: Always choose the most restrictive situations first.
a. the manager must be a woman
Choose the woman for the manager any of 6 ways.
That leaves 15 ways to choose the captain.
That leaves 14 ways to choose the assistant coach.
Answer: 6*15*14 = 1260 ways.
b. the captain must be robert (one of the men)
Choose Robert for the captain 1 way.
That leaves 15 ways to choose the manager
That leaves 14 ways to choose the assistant coach.
Answer: 1*15*14 = 210 ways.
c. at least one man and one women must be chosen for
the 3 distinct positions
We use the indirect method.
All choices - undesirable choices = desirable choices.
All choices:
Choose the captain any of 16 ways.
That leaves 15 ways to choose the manager
That leaves 14 ways to choose the assistant coach.
That's 16*15*14 = 3360 ways.
Undesirable choices of all three men:
Choose the captain any of 10 ways.
That leaves 9 ways to choose the manager
That leaves 8 ways to choose the assistant coach.
That's 10*9*8 = 720 ways to choose all men.
Undesirable choices of all three women:
Choose the captain any of 6 ways.
That leaves 5 ways to choose the manager
That leaves 4 ways to choose the assistant coach.
That's 6*5*4 = 120 ways to choose all women.
Answer: 3360 - 720 - 120 = 2520 desirable ways.
Edwin
