Questions on Algebra: Combinatorics and Permutations answered by real tutors!

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Question 167634: The neighborhood association with sixteen members also needs to select a committee of four to oversee fundraising. How many possible committees are there?: The neighborhood association with sixteen members also needs to select a committee of four to oversee fundraising. How many possible committees are there?
Answer by jim_thompson5910(9929) About Me  (Show Source):
You can put this solution on YOUR website!

Since order does not matter (ie no member on the committee is more important than the other), we must use the combination formula:


n!/(n-r)!r! Start with the combination formula



16!/(16-4)!4! Plug in n=16 and r=4



16!/12!4! Subtract 16-4 to get 12


Expand 16!
(16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/12!4!


Expand 12!
(16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/(12*11*10*9*8*7*6*5*4*3*2*1)4!



(16*15*14*13*cross(12*11*10*9*8*7*6*5*4*3*2*1))/(cross(12*11*10*9*8*7*6*5*4*3*2*1))4! Cancel



(16*15*14*13)/4! Simplify


Expand 4!
(16*15*14*13)/(4*3*2*1)



43680/(4*3*2*1) Multiply 16*15*14*13 to get 43,680



43680/24 Multiply 4*3*2*1 to get 24



1820 Now divide



So 16 choose 4 (where order doesn't matter) yields 1,820 unique combinations


So there are 1,820 ways to choose a committee of 4 members given a pool of 16 people.