A photographer arranges $6$ people in a row, but two specific persons refuse to stand next to each other. In how many ways can the arrangement be made?

To find the number of ways where two persons are not together, we can use the method:
$$\text{Required Ways}=\text{Total Arrangements}-\text{Arrangements where they are together}$$

Step 1: Calculate total ways to arrange $6$ people in a row:

$$\text{Total Ways}=6!=6\times 5\times 4\times 3\times 2\times 1=720$$

Step 2: Calculate ways where the two specific persons stand together:
Treat these two persons as a single block/unit. Now we have $(5-1)+1=5$ units to arrange.
Number of ways to arrange $5$ units = $5!=120$.
Within the block, the two persons can switch places in $2!=2$ ways.

$$\text{Ways Together}=5!\times 2!=120\times 2=240$$

Step 3: Calculate the required ways:

$$\text{Required Ways}=720-240=480$$