Variation of a $9$-digit string with digits from $0 - 9$ I've got an exercise where I have a $9$ digit string e.g. $123456789$ with every digit randomly between $0$ and $9$.
I want to find out how many combinations there are without repetition of whole string
For example:

*

*$123456789$

*$132456789$

*$213456789$

*...
should be counted as one because they have the same occurrence of numbers.

I've found this formula, but I don't think it's correct:
$${n+k - 1 \choose n}$$
If something is unclear just let me know and I'm happy for every help :)
EDIT:
Thank you all very much. It seems that
$${n+k - 1 \choose n}$$
was after all the right formula. To doublecheck I've implemented it in JS with a simple for loop, maybe someone wants to try it
const zeroPad = (num, places) => String(num).padStart(places, '0');

let count = [];
// pretty slow because of the big number
for (let i = 0; i <= 999999999; i++) {
  let padded = zeroPad(i, 9);
  padded = padded.split('').sort().join('');
  if (count.includes(padded)) {
    continue;
  } else {
    count.push(padded);
  }
}

console.log(count.length);

 A: From your description, it seems clear that it is combinations, not permutations that you are seeking.
Consider $10$ distinct bins numbered $0\; through\;9$ in which we are to put $9$ identical balls, with the balls getting labelled according to the bin in which they are put.
This is a classic stars and bars problem which will have the solution $$\binom{9+10-1}9 = 48620$$
A: What you are asking is how many nine-digit strings can be formed using the digits $0 - 9$ with repetition of digits permitted up to permutations of the digits.  Therefore, what matters here is how many times each digit appears in the string.  Thus, we seek the number of solutions of the equation
$$x_0 + x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 = 9 \tag{1}$$
in the nonnegative integers, where $x_i$ is the number of times the digit $i$ appears in the string, $0 \leq i \leq 9$.  Note that this is equal to the number of nine-digit strings with nondecreasing digits.
A particular solution of equation $1$ corresponds to the placement of nine sticks in a row of nine stones.  For instance,
$$| \bullet \bullet | | \bullet \bullet \bullet | | \bullet | | \bullet \bullet | | \bullet$$
corresponds to the nondecreasing string $113335779$ or, in your formulation, a string containing two $1$s, three $3$s, one $5$, two $7$s, and one $9$.
The number of such strings is the number of ways we can place nine sticks in a row of nine stones, which is
$$\binom{9 + 10 - 1}{10 - 1} = \binom{18}{9}$$
since we must choose which nine of the $18$ positions required for nine sticks and nine stones will be filled with sticks.  Alternatively, we can write this as
$$\binom{9 + 10 - 1}{9} = \binom{18}{9}$$
since we must choose which nine of the $18$ positions required for nine sticks and nine stones will be filled with stones.  The latter formulation corresponds to your formula
$$\binom{n + k - 1}{n}$$
if we take $n = 9$ to be the length of the string and $k = 10$ to be the number of types of objects which can be used in the string.
