Explanation:

Writing $60!$ in prime factorization format is difficult. So we only focus on the power of $5$ when $60!$ is written in prime factorization format.

$60! = 1 \times 2 \times 3 ...\times\boxed5$. . .$\times\boxed{10}...\times\boxed{25}$. . .$\times\boxed{50}$. . .$\times\boxed{60}$

$60! = 1 \times 2 \times 3 ...\times\boxed5$. . .$\times\boxed{2 \times 5}$. . .$\times\boxed{5^2}$. . .$\times\boxed{2\times5^2}$. . .$\times\boxed{5 \times 12}$

Every fifth number is a multiple of 5.

So there must be 60/5 = 12 fives are available.

In addition to this $25 = 5^2 $ and $50 = 2 \times 5^2$ contribute another two 5's. So total number is 12 + 2 = 14

$\left[ {\displaystyle\frac{{60}}{5}} \right] + \left[ {\displaystyle\frac{{60}}{{{5^2}}}} \right] = 12 + 2 = 14$

Here [ ] Indicates greatest integer function.

*Shortcut:*
Divide 60 by 5 and write quotient. Omit any remainders. Again divide the quotient by 5. Omit any remainder. Follow the procedure, till the quotient not divisible further. Add all the numbers below the given number. The result is the answer.