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

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- Aug 21st 2011, 07:17 AMhmmmm20 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, 09:12 AMTravellerRe: 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, 09:50 AMPlatoRe: 20 rolls of a die chance all numbers come up?
- Aug 21st 2011, 02:24 PMhmmmmRe: 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

.

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, 02:32 PMPlatoRe: 20 rolls of a die chance all numbers come up?
Very good: count the surjections. That is good.