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.