SOLUTION: The offices of the president, vice president, secretary, and treasurer for an environmental club will be filled from a pool of 15 candidates. nine members are apart of the debate t

Question 974097: The offices of the president, vice president, secretary, and treasurer for an environmental club will be filled from a pool of 15 candidates. nine members are apart of the debate team.
a.)what is the probability that all the offices are filled by members of the debate team?
b.)what is the probability that none of the offices are filled by members of the debate team?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
I'm going to use the Combination formula n C r = (n!)/(r!*(n-r)!)

a)

First compute 15 C 3 to figure out how many ways there are to pick 3 people from a pool of 15

n C r = (n!)/(r!(n-r)!)

15 C 3 = (15!)/(3!*(15-3)!)

15 C 3 = (15!)/(3!*12!)

15 C 3 = (15*14*13*12!)/(3!*12!)

15 C 3 = (15*14*13)/(3!)

15 C 3 = (15*14*13)/(3*2*1)

15 C 3 = (2730)/(6)

15 C 3 = 455

So there are 455 ways to pick 3 people from a pool of 15

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Now compute 9 C 3 to figure out how many ways there are to pick 3 people from just the debate team.

n C r = (n!)/(r!(n-r)!)

9 C 3 = (9!)/(3!*(9-3)!)

9 C 3 = (9!)/(3!*6!)

9 C 3 = (9*8*7*6!)/(3!*6!)

9 C 3 = (9*8*7)/(3!)

9 C 3 = (9*8*7)/(3*2*1)

9 C 3 = (504)/(6)

9 C 3 = 84

There are 84 ways to pick 3 people from a pool of 9 (the debate team only)

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Divide the results:

(9 C 3)/(15 C 3) = 84/455 = 0.184615

The probability of having all 3 members selected from the debate team is approximately 0.184615

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b)

There are 15 - 9 = 6 people that aren't on the debate team

Compute 6 C 3 to get...

n C r = (n!)/(r!(n-r)!)

6 C 3 = (6!)/(3!*(6-3)!)

6 C 3 = (6!)/(3!*3!)

6 C 3 = (6*5*4*3!)/(3!*3!)

6 C 3 = (6*5*4)/(3!)

6 C 3 = (6*5*4)/(3*2*1)

6 C 3 = (120)/(6)

6 C 3 = 20

(6 C 3)/(15 C 3) = 20/455 = 0.043956

The probability of having no members of the debate team selected is roughly 0.043956

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Jim
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