This is a binomial probability distribution. I'm going to use n for the number of digits guessed correctly out of m digits.

$p=Pr[$guessing a digit correctly$]=\dfrac 1 {10}$

$Pr[$guessing $\geq n$ correctly$]=1-Pr[$guessing $<n$ correctly$]$

$Pr[$guessing $k$ correctly$]=\begin{pmatrix}m \\ k \end{pmatrix}(1-p)^{n-k}p^k$

$Pr[$guessing $<n$ correctly$]=\displaystyle{\sum_{k=0}^{n-1}}\begin{pmatrix}m \\ k \end{pmatrix}(1-p)^{n-k}p^k$

Note that $\begin{pmatrix}m \\ k \end{pmatrix}$ is "m choose k", or the binomial coefficient(m,k) $=\dfrac {m!}{(m-k)!k!}$

so

$Pr[$guessing $\geq n$ correctly$]=1-\displaystyle{\sum_{k=0}^{n-1}}\begin{pmatrix}m \\ k \end{pmatrix}(1-p)^{n-k}p^k$