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Let $S$ be a set that includes $\{1,2,3,4,5,...,15\}$ We are picking $5$ numbers out of the set $S$ with replacement. What is the probability that their sum is $14$?

I know we can use the Sticks and Stones (Stars and bars) method but I'm a bit shaky on that.

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you need to find the number of solutions to $a_1+a_2+a_3+a_4+a_5=14$ such that each number is an element in $\{1,2,\dots,15\}$.

If we let $b_j=a_j-1$ it is clear that this is equal to the number of solutions to $b_1+b_2+b_3+b_4+b_5=9$ such that each number is a non-negative integer.

So we have to split $9$ stars by using $4$ bars. The answer is therefore $\binom{9+4}{4}=\binom{13}{4}$.

The total number of outcomes is $15^{5}$.

So the probability is $\dfrac{\binom{13}{4}}{15^5}$

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    why did you use 9? is it because you arbitrarily chose that $b_1 = (a_1 + 1)$? – 2017-01-25
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    finding $5$ positive integers that add up to $14$ is the same as finding $5$ non-negative integers that add up to $9$. – 2017-01-25
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    How about finding 5 non-negative integers that add up to 14? – 2017-01-25
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    that can be done in $\binom{14+4}{4}$ ways, because there are $14$ stars and $4$ bars. – 2017-01-25