Question 1205549: A club consists of 16 men and 17 women. In how many ways can they choose a president, vice president, treasurer, and secretary, along with an advisory committee of six people?
Answer by math_tutor2020(3816)   (Show Source):
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16 men + 17 women = 33 people total
33*32*31*30 = 982080 ways to pick the president, VP, treasurer and secretary. Order matters because the seats or positions have different names.
The basic idea is to start at 33 and count down by 1 until 4 slots are accounted for. We multiply each value along the way.
Alternatively, you can use the nPr formula with n = 33 and r = 4.
After 4 people are chosen for those named seats there are 33-4 = 29 people left and there are 6 committee seats to fill.
Unlike the previous 4 positions, order will not matter with these new 6 seats.
This is because each person on the 6-person committee is of equal rank.
Note how none of the 6 seats are named.
We have n = 29 people and r = 6 seats to fill.
Using the nCr combination formula we get:
n C r = (n!)/(r!(n-r)!)
29 C 6 = (29!)/(6!*(29-6)!)
29 C 6 = (29!)/(6!*23!)
29 C 6 = (29*28*27*26*25*24*23!)/(6!*23!)
29 C 6 = (29*28*27*26*25*24)/(6!)
29 C 6 = (29*28*27*26*25*24)/(6*5*4*3*2*1)
29 C 6 = 342014400/720
29 C 6 = 475020
Pay close attention to the fact "29*28*27*26*25*24" is in the numerator of that nCr calculation above. The "29*28*27*26*25*24" calculates the permutation npr(29,6). We divide by 6! = 6*5*4*3*2*1 = 720 because this is the number of ways to rearrange any group of 6 people.
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We found that...- 982080 ways to pick president, VP, treasurer and secretary (order matters)
- 475020 ways to pick the 6-person committee (order doesn't matter)
There will be 982080*475020 = 466,507,641,600 (roughly 466.5 billion) different ways to select all 10 people.
Effectively we just computed: npr(33,4)*ncr(29,6)
Alternatively you can compute it like this: ncr(33,6)*npr(27,4)
Why does this work? Because we can form the committee of 6 people first, and then we'd have 33-6 = 27 people left to pick from to select the president, VP, treasurer, and secretary.
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Answer: 466,507,641,600
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