$\prod _{ }^{ }{ }$ & $\sum { }$

We note that the RHS is the partial fractions of the LHS.

$\begin{aligned} \frac {25!}{\prod_{i=0}^{25} (x+i)} & = \sum_{i=0}^{25} \frac {A_i}{x+i} \\ 25! & = \sum_{i=0}^{25} A_i \prod_{k \ne i}^{25} (x+k) \quad \quad \small {\color{#3D99F6}\text{Putting }x = -24 \text{, all terms in the RHS with }(x+24) \text{ become }0 \text{ except 24th term.}} \\ 25! & = A_{24}(-24)(-23)(-22)...(-2)(-1)(1) \\ & = 24!A_{24} \\ \implies A_{24} & = \boxed{25} \end{aligned}$