// test
for (int i = 1; i <= 200; i++)
{
    Dictionary<int, int> change = MakeChange(new List<int> { 1, 5, 10, 25, 50 }, i);
    Debug.WriteLine($"sum {i}");
    int sum = 0;
    foreach (KeyValuePair<int, int> coins in change)
    {
        sum += coins.Key * coins.Value;
        Debug.WriteLine($"  coin {coins.Key}  count {coins.Value} ");
    }
    if (i != sum)
    {
        Debug.WriteLine("problem");
    }
}

// end test

private static Dictionary<int, int> MakeChange(List<int> coins, int sum)
{
    if(sum < 0 || coins.Count == 0)
    {
        throw new ArgumentOutOfRangeException();
    }
    Dictionary<int, int> change = new Dictionary<int, int>();
    foreach (int coin in coins.Distinct().Where(x => x > 0).OrderByDescending(x => x))
    {
        int j = sum / coin;  //integer math rounds down
        if (j > 0)
        {
            change.Add(coin, j);
        }
        sum -= j * coin;
        if (sum == 0)
            return change;
    }
    return null;
}