In how many different ways can the top eight new indie bands be ranked on a top eight list? The top hit song for each of the eight bands will compete to receive monetary awards of $1000, $500 and $250, respectively. In how many ways can the awards be given out?
I think the first question is 8 different ways because they will be ranked on a list of 8.
I think the second question is 8*8*8=512 ways the awards can be given out.....but im not sure if i did this right, please help!
say we have 8 bands numbered 1 through 8
chance that band 1 is top = 1/8
then chance that band 2 is second = 1/7
then chance that band 3 is third = 1/6
etc
so I claim
8! ways (big number, you compute :)
Now how many permutations of 8 taken 3 at a time (permutations not combinations because the order of prizes counts within each group of three) ?
n!/(n-r)! = 8!/(8-3)! = 8!/5! = 8*7*6
= 336
oh okay i get it...
You are correct in thinking that there are 8 different ways the top eight new indie bands can be ranked on a top eight list. Since no two bands can be ranked in the same position, each band has 8 possible positions they can be ranked in.
To calculate the number of ways the awards can be given out, we need to consider the number of choices for each award position.
For the first award of $1000, any of the 8 bands can receive it. So there are 8 choices for this position.
For the second award of $500, after one band has already received the $1000 award, there remain 7 bands to choose from. So there are 7 choices for this position.
For the third award of $250, after two bands have already received awards, there remain 6 bands to choose from. So there are 6 choices for this position.
To calculate the total number of ways the awards can be given out, we multiply the number of choices for each position: 8 * 7 * 6 = 336.
Therefore, there are 336 different ways the awards can be given out to the top eight new indie bands.