Question 1022266: Please help me with this problem!
A club has 43 members, 19 are age greater than or equal
to 30 years old and 24 are age less than 30 years old.
A slate of 6 officers will be elected including a
president, vice president, treasury, Sargent of arms
and press liaison.
How many slates of 6 officers are there in which the
president and vice president are not from the same age
group. (Helpful hint: One is age greater than equal to
30 years old and one is less than 30 years old)
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website! Please help me with this problem!
A club has 43 members, 19 are age greater than or equal
to 30 years old and 24 are age less than 30 years old.
A slate of 6 officers will be elected including a
president, vice president, secretary, treasurer,
sergeant-at-arms and press liaison.
How many slates of 6 officers are there in which the
president and vice president are not from the same age
group. (Helpful hint: One is age greater than equal to
30 years old and one is less than 30 years old).
First we find the number of ways to choose the top
pair of officers:
Case 1: The president is 30 or older and the
vice-president is younger than 30.
19 ways to choose the president
24 ways to choose the vice-president
That's 19*24 = 456 ways to have the president older
and the vice-president younger
Case 2: The president is younger than 30 and the
vice-president is 30 or older.
24 ways to choose the president
19 ways to choose the vice-president
That's also 24*19 = 456 ways to have the president
younger and the vice-president older.
So there are 2*456 or 912 ways to choose the pair
of top officers.
For each of these 912 choices for the pair of top
officers, there remain:
41 ways to choose the secretary.
40 ways to choose the treasurer.
39 ways to choose the sergeant-at-arms.
38 ways to choose the press liason.
That's 912*41*40*39*38 = 2216597760 ways.
Edwin
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