SOLUTION: If I roll 5 regular dice out of a cup, what is 1. The probability that I will get 5 of a kind. I think the answer is 1 in 1296. 2. What is the probability that I will get two p

There are 6 ways to succeed. They are to get 11111,22222,...,66666

Now we figure the number of ways they can land.

Imagine the 5 dice being labeled #1,#2,#3,#4,#5.

Die #1 can land any of 6 ways.
Die #2 can land any of 6 ways
Die #3 can land any of 6 ways.
Die #4 can land any of 6 ways.
Die #5 can land any of 6 ways.

That's 6 ways out of 6∙6∙6∙6∙6 = 7776 or 6/7776 = 1/1296

Another way to look at it is this:

Regardless of what number die A lands on,
the probability that die B lands on that number is 1/6.
the probability that die C lands on that number is 1/6.
the probability that die D lands on that number is 1/6.
the probability that die E lands on that number is 1/6.

(1/6)(1/6)(1/6)(1/6) = (1/6)(4 = 1/1296

You are right!

Case 1. There are 4 the same and one with a different number. (like 33433)

Choose the number for there to be 4 of in 6 ways
Choose the one number to be different in 5 ways.
Choose the die to get the one different number in 5 ways.
All the others can land on the other number in 1 way.

That's 6∙5∙5∙1 = 150 ways

Case 2. The two pairs are different numbers and one that's different from either
(like 22335)

Choose the two numbers for the pairs in 6C2 = 15 ways.
Choose the 2 dice to get the smaller of those two numbers in 5C2 = 10 ways.
Choose the 2 of the 3 remaining dice as the pair to get the larger of those two
numbers in 3C2 = 3 ways.
Choose the 1 remaining die to get the single number in only 1 way.

That's 15∙10∙3∙1 = 450 ways.

Total number of successful ways = 150+450 = 600 ways

We have shown in the first problem that the total number of ways the 5 dice can
land is 7776, so the desired probability is 600/7776 = 24/324.

Edwin