Vtyjkyuk

I want to check whether my concept and answer is right

I am considering strings of four decimal digits that contain the same digit twice. With this,
I have the possibilities of $xxyz,xyxz,yxxz,yxzx,yzxx,xyzx$ where $x$ is the same digit, and $y, z$ are randomly different decimal digits taken.
So $10 cdot 1 cdot 9 cdot 8 = 720$ and $6$ possibilities, then $720 cdot 6 = 4320$ ways.
Now consider digits that have pattern $xxyy, yyxx,xyxy,yxxy,yxyx,xyyx$, here $10 cdot 1 cdot 9 cdot 9=810 cdot 6= 4860$ ways.
Then total combinations are $10^4 =10000$ then $10000-4320-4860=820$ ways .

Your title is ambiguous (see the comment of Mohammad) and I preassume that it concerns the number of strings that have the property that no digit appears exactly $2$ times.

Where it concerns strings that have $3$ distinct digits your calculation with outcome $$binom42cdot10cdot9cdot8=4320$$ is correct.

Where it concerns strings that have $2$ distinct digits that both appear twice the outcome should be: $$frac12binom42cdot10cdot9=270$$
You (maybe accidently) have $2$ factors $9$ instead of $1$ and did not repair double counting.

You could also reason that there are not $6$ but $3$ patterns there ($xxyy$,$xyxy$ and $xyyx$) and this with $10$ choices for $x$ and $9$ remaining choices for $y$.