Simple Question About RNG

First of all, true randomness does not guarantee that two numbers with the same value never occur in a sequence.
For example, throwing a die is considered to be random, but you would not be shocked if you get three sixes in a row, right.

That said, the solution to your problem is to store your numbers in a list, generate a new number and check if it exists in the list.
If it does, generate a new number and check again.

List<int> randomNumbers = new List<int>();
Random randy = new Random();
while (randomNumbers.Count < 10)
{
    int randomNumber = randy.Next(1,10);
    if (!randomNumbers.Contains(randomNumber))
        randomNumbers.Add(randomNumber);
}