Can a Monkey Rearrange Letters to Spell Illinois and Tennessee?

[uiuc2014]

I just meet some easy problem like this,
Does a monkey have a better chance of rearranging
or
I I I L L N O S to spell I L L I N O I S
E E E E N N S S T to spell T E N N E S S E E ?

For ILLINOIS
there are 3"I",2"L",1"N",1"O",1"S".
So P(Illinois)= 8!/(3!*2!*1!*1!*1!)
My question is that why the answer of ( 8 )*( 5 )*( 3 )*( 2 )*( 1 )
////////////////////////////////////////////( 3 )//( 2)//( 1 )//( 1 )/( 1 )

is exactly 2 times larger than the former one?


(8)
(3)means selecting 3 out of 8

 

[HallsofIvy]

I just meet some easy problem like this,
Does a monkey have a better chance of rearranging
or
I I I L L N O S to spell I L L I N O I S
E E E E N N S S T to spell T E N N E S S E E ?

For ILLINOIS
there are 3"I",2"L",1"N",1"O",1"S".
So P(Illinois)= 8!/(3!*2!*1!*1!*1!)
My question is that why the answer of ( 8 )*( 5 )*( 3 )*( 2 )*( 1 )
////////////////////////////////////////////( 3 )//( 2)//( 1 )//( 1 )/( 1 )

is exactly 2 times larger than the former one?

What "former one" are you talking about?
$$\frac{8!}{3! 2!}= \frac{8(7)(6)(5)(4)(3!)}{3! 2!}$$
and we can cancel the "3!" in numerator and denominator. That leaves
$$\frac{8(7)(6)(5)(4)}{2}= 3360$$

If you are saying that the "answer" (to what question) is
$$\frac{8(5)(3)(2)(1)}{3(2)}= 8(5)= 40$$
neither is twice the other.

(8)
(3)means selecting 3 out of 8

But what does that have to do with this question?