# 20 rolls of a die chance all numbers come up?

• Aug 21st 2011, 06:17 AM
hmmmm
20 rolls of a die chance all numbers come up?
If we roll a die 20 times what are the chances that all numbers 1 through 6 come up?

I thought the size of the space was 6^20 and then the size of the event would be
(6!*6^12) however this is obviously wrong.

Thanks for any help
• Aug 21st 2011, 08:12 AM
Traveller
Re: 20 rolls of a die chance all numbers come up?
Can you compute the chances that at least one number does not come up, using derangements?
• Aug 21st 2011, 08:50 AM
Plato
Re: 20 rolls of a die chance all numbers come up?
Quote:

Originally Posted by hmmmm
If we roll a die 20 times what are the chances that all numbers 1 through 6 come up?

I think that reply #2 means use inclusion/exclusion.
Lets play the "back-of-the-book" game.

You look up the answer and see $\sum\limits_{k = 0}^5 {\left( { - 1} \right)^k \binom{6}{k}\left( {\frac{{6 - k}}{6}} \right)^{20} } .$
WHAT THE HECK????
Can you tell us all what is going on there?
• Aug 21st 2011, 01:24 PM
hmmmm
Re: 20 rolls of a die chance all numbers come up?
Ah ok so If it as all the ways you can choose the 20 with all 6 die thrown then that is the number of surjections from the set of 6 elements to the set of 20 elements which is

$\Sigma^{5}_0 (-1)^k 6Ck (6-k)^{20}$.

We then simply divide by the total space 6^20 and arrive at what you have there.

Sorry quite simple I was just thinking of it in a completely and much less helpful way.

thanks a lot (sorry for so much editing I made loads of typo's)
• Aug 21st 2011, 01:32 PM
Plato
Re: 20 rolls of a die chance all numbers come up?
Very good: count the surjections. That is good.