1. The problem is to evaluate the hypergeometric function $_2F_1(10,7;6;z)$ at a specific value of $z$, but since $z$ is not provided, we assume the general form. 2. The hypergeometric function $_2F_1(a,b;c;z)$ is defined by the series: $$ _2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!} $$ where $(q)_n$ is the Pochhammer symbol (rising factorial). 3. Here, $a=10$, $b=7$, $c=6$. Note that $c=6$ is less than $a$ and $b$, and the series converges for $|z|<1$. 4. Without a specific $z$ value, the function cannot be numerically evaluated. 5. If you want to evaluate $_2F_1(10,7;6;z)$ for a particular $z$, please provide $z$. Final answer: The hypergeometric function $_2F_1(10,7;6;z)$ is defined by the series above but cannot be evaluated without a value for $z$.