Answer to Question #26575 in Combinatorics | Number Theory for Paul

Question

There are 27 members of a committee. What are the odds of guessing the 14 members of the committee that attended a meeting? How does one calculate the result ?

Answer 1

We have that the committee consists of 27 members and 14 of them are randomly chosen to attend the meeting.

Recall that the number of choices of n elements from k is called the binomial coefficient
C_n^k and it is equal to C_n^k = n! / (k! * (n-k)! ), where n! =n*(n-1)*(n-2)*...*2*1.
For instance, the number of choices of 2 elements from 5 is equal
C_5^2 = 5! /(2! * 3!) = 5*4*3*2*1 / ( (2*1) * (3*2*1) ) = 5*4/(2*1) = 20/2 = 10.
If we have the set {1,2,3,4,5} then the choices of 2 elements are the following 10 ones:
(1,2), (1,3), (1,4), (1,5)
(2,3), (2,4), (1,5)
(3,4), (3,5),
(4,5)
The number of choices of 14 members from 27 is equal to the binomial coefficient C_{27}^{14} =27!/(14! * (27-14)!).
Therefore the odds of guessing the 14 members of the committee from 27 ones that attended a meeting is equal to 1/C_{27}^{14}.
Let us compute that number:
C_{27}^{14} = 27!/(14! * (27-14)!) = 27!/(14! * 13!) = 27*26*25*24*23*22*21*20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2:(14*13*12*11*10*9*8*7*6*5*4*3*2 * 13*12*11*10*9*8*7*6*5*4*3*2) = 20058300.
Hence the odds of guessing the 14 members from 27 ones is 1/20058300.