Question 952834
16 people ( 10 men & 6 women) are on a team. How many different 
ways can the team chose a captain, manager, assistant coach if:
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Rule:  Always choose the most restrictive situations first.
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a. the manager must be a woman
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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.
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b. the captain must be robert (one of the men) 
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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.
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c. at least one man and one women must be chosen for 
the 3 distinct positions
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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</pre>