# In how many ways....

#### aburchett

##### New member
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, $250 and$100, respectively. In how many ways can the awards be given out?

My work:
In how many different ways can the top eight new indie bands be ranked on a top eight list?
P[sub:16992ywf]8[/sub:16992ywf][sub:16992ywf],[/sub:16992ywf][sub:16992ywf]8[/sub:16992ywf] = 8!/(8-8)! = 8!/0! = 8*7*6*5*4*3*2*1/1 = 40,320
The top eight new indie bands can be ranked on a top eight list 40,320 different ways.

In how many ways can the awards be given out?
8_C_4 = 8!/(8-4)! = 8!/4! = 8*7*6*5*4!/4! = 1,680
The awards can be given out 1,680 different ways.

Is this correct?

#### soroban

##### Elite Member
Hello, aburchett!

Correct answers . . . just one typo.

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, $250 and$100, respectively.
In how many ways can the awards be given out?

My work:

In how many different ways can the top eight new indie bands be ranked on a top eight list?

$$\displaystyle P(8,8) \,=\, \text{8!}{0!} \,=\,\frac{ 8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{1} \:=\: 40,320$$

The top eight new indie bands can be ranked on a Top Eight list on 40,320 different ways. .
Right!

In how many ways can the awards be given out?

$$\displaystyle _8C_4 \:=\: \frac{8!}{(8-4)!} \:=\: \frac{8!}{4!} \:=\: \frac{8\cdot7\cdot6\cdot5\cdot4!}{4!} \:=\: 1,680$$
. $$\displaystyle \uparrow$$

You mean P

The awards can be given out 1,680 different ways. .
Yes!