Potato, Optato

We can arrange the letters in $6!=720$ ways when the letters are distinct. But since letter 't' and letter 'o' appear twice, the number of ways would be $\dfrac{ 6!}{ 2!\times2!} = \dfrac{720}4 = \boxed{180}$ .

We have six places, one for each letter. In the first, we can put $6$ different letters, in the next we can only put $6-1=5$ letters, in the next one, $6-2=4$ letters and so on. Hence, we can arrange the letters in $6·5·4·3·2·1=6!=720$ ways.

But, we have $2$ o's and $2$ t's. So, all the possible words will be duplicated because of the $2!=2$ o's and duplicated again because of the $2!=2$ t's.

Thus the solution is the total number of ways we can arrange the letters divided by the number of duplications:

$\frac { 6! }{ 2!2! } =\frac { 720 }{ 2·2 } =\boxed { 180 }$ .

Note that the word $\text{Potato}$ has $6$ letters, with $\text{o}$ and $\text{t}$ each appearing twice. Hence the number of ways the letters in the word $\text{Potato}$ can be arranged is $\frac{6!}{2!2!}=\boxed{180},$ and we are done. $\square$

$\Rightarrow \frac { (total\quad number\quad of\quad letters)! }{ (number\quad of\quad repeats)! } \\ \Rightarrow \frac { 6! }{ 2!\quad \cdot \quad 2! } \\ \Rightarrow 180$

Since only two letters are distinct we have 6 places to put the first letter, and 6-1 =5 places to put the second letter. Now there are four places left and there are two of each letter left. So there are 4 choose 2 places to put the next two letters of the same type and the remaining places are the last two letters of the same type. Therefore the answer is 6 *5 * (4 choose 2).

"Potato" has 6 letters, so these can be arranged in 6 * 5 * 4 * 3 * 2 * 1 = 6! ways.
But in "potato", 't' occurs 2 times, so it can be arranged in 2! ways.
Same for 'o'.
Now calculate the answer as 6!/(2! * 2!) = 720/(2 * 2) = 180

Answer is 6! divided by $2! \times 2!$ because there is a total of 6 digits in ''potato'' and ''o'' and ''t'' are appearing twice.