Example 1
seven speed skaters are competing in an olympic race. the first-place skater earns the gold medal, the second-place skater earns the silver medal, and the third-place skater earns the
bronze medal. in how many different ways could the gold, silver, and bronze medals be awarded? the letters a, b, c, d, e, f, and g will be used to represent these seven skaters.
how can we determine the number of different possible outcomes? how many are there?
now consider a slightly different situation. seven speed skaters are competing in an olympic race. the top three skaters move on to the next round of races. how many different
"top three" groups can be selected?
how is this situation different from the first situation? would you expect more or fewer possibilities in this situation? why?
would you consider the outcome where skaters b, c, and a advance to the final to be a different outcome from a, b, and c advancing?
a permutation is an ordered arrangement (a sequence) of items from a set of nn distinct items.
in contrast, a combination is an unordered collection (a set) of items from a set of nn distinct items.
when we wanted to know how many ways there are for seven skaters to finish first, second, and third, order was important. this is an example of a permutation of 3 selected from
a set of 7. if we want to know how many possibilities there are for which three skaters will advance to the finals, order is not important. this is an example of a combination of 3 selected
from a set of 7.