The number of ways one can select three cards from a group of n cards (assuming the order of the selection matters), where , is given by . For a certain card trick a magician has determined that there are exactly 5814 ways to choose three cards from a given group. How many cards are in the group?
A)18 cards
B)17 cards
C)19 cards
D)20 cards
E)21 cards

Question

The number of ways one can select three cards from a group of n cards (assuming the order of the selection matters), where  , is given by  . For a certain card trick a magician has determined that there are exactly 5814 ways to choose three cards from a given group. How many cards are in the group?
A)18 cards 
B)17 cards 
C)19 cards 
D)20 cards 
E)21 cards

Quizplus · Accepted Answer

We need to solve the equation 11ebfdc1_ec28_6dac_abe3_add58abfc852_TBX8727_11 = 5814. We can simplify the equation as (n)(n-1)(n-2)/3! = 5814. Rearranging, we get n^3 - 3n^2 + 2n - (3)(5814) = 0. We can use trial and error to find that n = 19 is a solution to this equation. Therefore, the group has 19 cards (option C).

The Number of Ways One Can Select Three Cards from a Group