SOLUTION: An elected official wants to take five members of his staff to an undisclosed secure location. What is the minimum number of staff members the elected official must have in order t

Question 1190190: An elected official wants to take five members of his staff to an undisclosed secure location. What is the minimum number of staff members the elected official must have in order to have at least 20 different groups from which to choose?
A)7 B)8 C)9 D)10 E)11
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

n = number of total staff members
r = number of staff members selected to go to the undisclosed secure location

We're told that r = 5, but we don't know what n is.
The list of possible options are: 7, 8, 9, 10, 11
which are drawn from choices A through E.

If n = 7, then,
C(n,r) = (n!)/(r!(n-r)!) .............. nCr combination formula
C(7,5) = (7!)/(5!*(7-5)!)
C(7,5) = (7!)/(5!*2!)
C(7,5) = (7*6*5*4*3*2*1)/((5*4*3*2*1)*(2*1))
C(7,5) = (5040)/((120)*(2))
C(7,5) = (5040)/(240)
C(7,5) = 21

I used the nCr combination formula because order doesn't matter.

We see that n = 7 and r = 5 makes 21 different groups, which fits the criteria of "at least 20" (aka "20 or more").

Using larger values of n will make C(n,r) larger than 21.
You can use Pascal's Triangle as an alternative route to computing the nCr values.

Answer: A) 7

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