SOLUTION: A 40-member club will elect a president and elect a vice president. How many possible ways can a president and vice president be elected from this club/ Suppose the 40 member c

Algebra ->  Permutations -> SOLUTION: A 40-member club will elect a president and elect a vice president. How many possible ways can a president and vice president be elected from this club/ Suppose the 40 member c      Log On


   



Question 626167: A 40-member club will elect a president and elect a vice president. How many possible ways can a president and vice president be elected from this club/
Suppose the 40 member club instead elects two co-presidents. How many possible ways can co-presidents be elected from this club?
How are these questions alike and how are they different?

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
n = 40
want to elect a president and vice president.
formula is P(40,2)
P is for permutation with n = 40 and x = 2.
the number of possible permutation is equal to 40! / (38!) which becomes:
40 * 39 = 1560

n = 40
want to elect 2 co-presidents.
formuls is C(40,2)
C is for combination with n = 40 and x = 2.
the number of possible combinations is equal to 40! / (38! * 2!) which becomes:
40 * 39 / 2 = 1560 / 2 = 780

the difference is in the first case order was important and in the second case order was not.

this is easier to see if n = 3.
you want to elect a president and vice president.
the formuls now becomes P(3,2) which becomes 3! / 1! which becomes 3*2 = 6 possible ways.
those ways are:
president     vice president
     a             b
     a             c
     b             c
     b             a
     c             a
     c             b

order was important because candidate A could be president or vice president.
now you want to elect 2 co-presidents.
the formula now becomes C(3,2) which becomes 3! / (1! * 2!) becomes 3*2/2 = 3
those ways are:
co-president     co-president
     a               b
     a               c
     b               c

order was not important here since there is no difference between a co-president and a co-president
candidate A can be one of the candidates for co-president while candidate B can be one of the candidates for co-president - which co-president position each takes doesn't matter.