Correctly predicting salt and sugar in the proper sequence. We have $10$ canisters, $5$ containing sugar and $5$ containing salt. What is the probability of naming them all in the correct order? (example: salt, sugar, sugar, salt, sugar, salt, salt, ...) 
It's different from predicting $10$ coins head or tails, because there are limited quantities of each.
Thanks.
 A: You have $\dbinom{10}{5}=252$ different options, so the probability is $\dfrac{1}{252}$
A: An easy way to look at this problem is to first simplify it by assuming there are only $4$ canisters.  Let A = salt and B = sugar.  Then the possible outcomes are AABB, ABAB, ABBA, BAAB, BABA, BBAA.  To determine the number of arrangements of A and B so there that are exactly $2$ of each, just "place" the $2$ As and the $2$ Bs must fill in the other spots.  There are $4 \choose 2$ = $6$ ways to place the $2$ As.  So if we were to guess the order for this simplified example, assuming we remembered to guess a pair of As and a pair of Bs, we would have a $1$ in $6$ chance of guessing correctly.  For $10$ canisters the logic/method is the same except it would be ${10 \choose 5}^{-1}$ = $1$ / $252$ = about $0.3968254$% probability.
Yes this is different than predicting $10$ fair coin flips but if you first told us that you got $5$ heads and $5$ tails, then it would be the same as this sugar/salt problem.
A: If we assume that each arrangement of $5$ salt and $5$ sugar canisters are equally likely, then any given arrangement is uniquely determined by the placement of, say, the $5$ sugar canisters among the $10$ canisters total.  For example, we can describe one arrangement as $(2, 3, 5, 7, 10)$, meaning that the sugar canisters are found in positions $2$, $3$, $5$, $7$, $10$ (and the salt canisters are in positions $1$, $4$, $6$, $8$, $9$).  Then clearly, there are exactly $$\binom{10}{5} = 252$$ ways to select $5$ distinct numbers from the set $\{1, 2, \ldots, 10\}$, representing the positions of the sugar canisters, and since each such selection is equally likely (by assumption), the probability of randomly choosing the unique correct representation is $1/252$.
