Arranging items in strictly increasing order "We throw 3 dice one by one. What is the probability that we obtain 3 points in strictly increasing order"?
Isn't the answer just $1/6$ as there are $3!$ possible permutations and only 1 permutation in which the 3 dice will be in strictly increasing order. However, the answer in my book is $5/54$.
Similarly, for the question of "what is the probability of 10 people being seated in strictly increasing order of age at a table?" Wouldn't the solution be $20/10!$ as direction of the increasing order has not been specified. I think this answer is correct whilst my answer for the previous question is wrong even though I am using the same logic and I can't see why this is, when they are effectively very similar questions. 
Thank You
 A: For a strictly increasing sequence, you need three different numbers to show, and each such outcome has just one of strictly increasing sequence, hence
$Pr = \dfrac{\binom{6}{3}}{6^3}= \dfrac{20}{216}= \dfrac{5}{54}$
PS
I believe your answer for the 2nd question is correct for a round table. For seating in a row (assuming either direction) it would be $\dfrac{2}{10!}$  
A: For each selection of $3$ out of $6$ numbers, there is exactly one way to arrange them in order, so there are $\binom63=20$ different strictly ordered outcomes, which yields a probability of $20/6^3=5/54$.
For the seating question, your answer is correct if this is a round table and the we can start anywhere and move in either direction. The difference to the first case is that you're placing exactly those $10$ people, whereas in the case of the dice you can select some elements out of a larger set and then arrange them in order.
A: The possible arrangements that make that happen, since the order counts, are
$123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456$ , so there's $20$ of them.
How many arrangements are possible? Well, $6^3$ is the right answer so it's $216$.
$$\frac{20}{216} = \frac{5}{54}$$
A: For 2) it is $\frac{1}{10!}$ assuming all people are of different age
