Trailing Zeros in the Product of the first 50 factorials

Each multiple of $5$ adds one trailing zero, each multiple of $10$ adds one trailing zero and each multiple of $25$ adds two trailing zeroes.

• Factorials from $5$ to $9$ are multiples of $5$ , so 1 trailing zero for each one of them, totalizing $5$ .
• Factorials from $10$ to $14$ are multiples of $5$ and $10$ , so 2 trailing zeroes for each one of them, totalizing $10$ .
• Factorials from $15$ to $19$ are multiples of $5$ , $10$ and $15$ , so 3 trailing zeroes for each one of them, totalizing $15$ .
• Factorials from $20$ to $24$ are multiples of $5$ , $10$ , $15$ and $20$ , so 4 trailing zeroes for each one of them, totalizing $20$ .
• Factorials from $25$ to $29$ are multiples of $5$ , $10$ , $15$ , $20$ and $25$ , so 6 trailing zeroes for each one of them, totalizing $30$ .
• Factorials from $30$ to $34$ are multiples of $5$ , $10$ , $15$ , $20$ , $25$ and $30$ , so 7 trailing zeroes for each one of them, totalizing $35$ .
• Factorials from $35$ to $39$ are multiples of $5$ , $10$ , $15$ , $20$ , $25$ , $30$ and $35$ , so 8 trailing zeroes for each one of them, totalizing $40$ .
• Factorials from $40$ to $44$ are multiples of $5$ , $10$ , $15$ , $20$ , $25$ , $30$ , $35$ and $40$ , so 9 trailing zeroes for each one of them, totalizing $45$ .
• Factorials from $45$ to $49$ are multiples of $5$ , $10$ , $15$ , $20$ , $25$ , $30$ , $35$ , $40$ and $45$ , so 10 trailing zeroes for each one of them, totalizing $50$ .
• Factorial of $50$ is multiple of $5$ , $10$ , $15$ , $20$ , $25$ , $30$ , $35$ , $40$ , $45$ and $50$ , so $12$ trailing zeroes for it.

So our total is $\color{#3D99F6} 5 + 10 + 15 + 20 + 30 + 35 + 40 + 45 + 50 + 12 = \fbox{262}$