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Relevant wiki: Trailing Number of Zeros

$\large \displaystyle \text{To find the trailing zeroes we use a simple method.}$

$\large \displaystyle f(100000) = \color{#1E93A5}{\sum_{i=1}^k \left\lfloor{\frac{100000}{5^i}}\right\rfloor}$

$\large \displaystyle \left\lfloor\frac{100000}{5}\right\rfloor = \color{#D61F06}{20000}$

$\large \displaystyle \left\lfloor\frac{100000}{5^2}\right\rfloor = \color{#3D99F6}{4000}$

$\large \displaystyle \left\lfloor\frac{100000}{5^3}\right\rfloor = \color{#20A900}{800}$

$\large \displaystyle \left\lfloor\frac{100000}{5^4}\right\rfloor = \color{#69047E}{160}$

$\large \displaystyle \left\lfloor\frac{100000}{5^5}\right\rfloor = \color{magenta}{32}$

$\large \displaystyle \left\lfloor\frac{100000}{5^6}\right\rfloor = \left\lfloor\color{#EC7300}{6.4}\right\rfloor = \color{#EC7300}{6}$

$\large \displaystyle \left\lfloor\frac{100000}{5^7}\right\rfloor = \left\lfloor\color{#624F41}{1.28}\right\rfloor = \color{#624F41}{1}$

$\large \displaystyle \left\lfloor\frac{100000}{5^8} \right\rfloor = \color{cyan}{0}$

$\large \displaystyle \text{So total number of trailing zeroes} = \color{#D61F06}{20000} + \color{#3D99F6}{4000} + \color{#20A900}{800} + \color{#69047E}{160} + \color{magenta}{32} + \color{#EC7300}{6} + \color{#624F41}{1} = \color{royalblue}{\boxed{24999}}.$

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