How many ways are there to select a first-prize winner, a second-prize winner, and a third-prize winner from 100 different people who have entered a contest ?
This question is easy to solve as 1st prize can be given in 100 ways, then 2nd prize can be given in 99 ways and similarly, 3rd prize can be given in 98 ways, so total ways are => 100 * 99 * 98 => 970200 ways
But 100 * 99 * 98 = P(100,3) and since it is a permutation. ordering must be there but I don't see any ordering here.
Can anyone explain that what am I missing ?
Best Answer
The order is there because you have reasoned that there are:
The order in which you then calculate the product of these three numbers is of course, due to commutativity, not important. But their values result from taking the order into account.
Without taking order into account, you would simply need to pick 3 people out of 100 and that's a combination of 3 out of 100: $${100 \choose 3} = 161\,700$$ It's no coincidence that you find precisely $\tfrac{1}{6}$th of the answer with the permutation because there are $3!=6$ ways to order these 3 randomly chosen people into the three prize winning positions.