Question

There are seven ice skaters in a competition. How many different possibilities are there for the top three places? (A different ordering of the same three skaters counts as a different possibility.)

Answer 1

210 possibilities, if I am not mistaken.

I did a tree diagram. You first name the seven skaters (skater 1, skater 2, skater 3, ect..) and write them in a row.

After you do that, do 6 arrows from each number because you cannot have skater 1 be first and second place, same with 2 and 3 and so on, so you need only 6 arrows. Make sure when you label the arrows that none of the arrows are labeled as the first skater number you drew in the beginning.

Finally, from each arrow you just drew, draw 5 arrows labeled each skater EXCEPT the skater you drew in the beginning and the second skater.

To save time, you could just do all of the steps above for only one beginning skater. You don't have to do it to all of them, you would run out of paper and it would be super messy.

Count all of the outside arrows. You should have 30. Then, since there are 7 skaters and you did only one, do 30x7, which gives you the total outcomes of 210 possibilities.