yahtzee problem! due in 9 hours!

the problem is the following:

In the game of Yahtzee (or Yacht), five dice are thrown. Show that the probability of throwing a large straight (5 numbers in a row, the order does not matter) is 5/162 .
Alternatively, you may solve the following, harder, problem. Show that the probability of throwing a small straight (4 numbers in a row) is 10/81 . Do not count small straights which are also large straights.

so i did the first part as follows :
the 5 numbers in a row can be wether 65432 or 54321 thus the chances to get them will be : 2(5*4*3*2*1)/6^5 but i don't really know how to represent my answer in a professional way !! for the second one, i got no clue how to do it !!

ps: we throuw the dices only once not like in the real game when we throw them twice (i think).
thank you for your very needed help

Quote:

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Originally Posted by yaszine

the problem is the following:

In the game of Yahtzee (or Yacht), five dice are thrown. Show that the probability of throwing a large straight (5 numbers in a row, the order does not matter) is 5/162 .
Alternatively, you may solve the following, harder, problem. Show that the probability of throwing a small straight (4 numbers in a row) is 10/81 . Do not count small straights which are also large straights.

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Divide thy problems into two problems.

1)You get 1,2,3,4,5
2)You get 2,3,4,5,6

Without loss of generality the probability is the same in both problems.

If you are trying to get: 1,2,3,4,5.
So the probablity in that order is,

Multipled by the number of ways you can get these arrangements which is
Thus,


Now the probablity of: 2,3,4,5,6
Is too

Thus, the total probability is,