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
----------------------
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)
----------------------
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
-------------------------------------------------------
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
------------------------------------------------------------------------------------------------------------------------
If you need more one-on-one help, email me at jim_thompson5910@hotmail.com. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).
Alternatively, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Any amount is greatly appreciated as it helps me a lot. This donation is to support free tutoring. Thank you.
Jim
------------------------------------------------------------------------------------------------------------------------
|
|
|
