Solution:

Number the stones $1$ through $15$ in order. We note that the condition is equivalent to stipulating that the stones have either all odd numbers or all even numbers. There are $\binom{8}{5}$ ways to choose $5$ odd-numbered stones, and $\binom{7}{5}$ ways to choose all even-numbered stones, so the total number of ways to pick the stones is $\binom{8}{5} + \binom{7}{5} = 77$. $\binom{n}{k}$ is the number of ways to choose $k$ out of $n$ items. It equals $\frac{n!}{k!(n-k)!}$.