a) How many times must a die be thrown to be sure that the same number occurs twice?

 

b) How many times must two die be thrown to be sure that the same total score occurs at least six times?

 

c) How many times must n dice be thrown to be sure that the same total score occurs at least p times?

The answer to all three is infinite. While the chances are incredibly small, the chances are non-zero that every single roll (of 1 die, or multiple dice) will be exactly the same. Similarly, the chances are small that the same number will never come back up, but it is non-zero.

 

What you should be asking are, how many throws will be necessary to expect that the same number comes up twice. This is a very important concept in probability, the expected value, your book should have lots of information on it.

 

and, since this is homework help, what have you done towards calculating this on your own? We (forum members) will help where you get stuck, but we won't do your problems for you.