Stacks: Balanced Brackets Hacker Rank
Let's look at the code first

List&lt;string&gt; list = new List&lt;string&gt; { "({[]})", "][(]}})(" };



Poor choice of variable name. You picked expression just a little below, that is a good one. Why not use the plural form of it: expressions?




foreach (var barcketsList in list)



barcketsList contains a typo, not a big deal usually. However, you should pay some more attention for an interview.
bracket-s + List you really don't want hinting this is an array/collection/list twice.




foreach (var barcketsList in list)
// &lt;- unnecessary white line
{



Don't leave white line just anywhere within a method in your code. They should be used to divide the logical blocks, eg: declaration/initialization vs logic vs return, logic#1 vs logic#2




string expression = barcketsList;
Stack&lt;char&gt; stack = new Stack&lt;char&gt;();
char[] symbols = expression.ToCharArray();



A temporary variable expression is not necessary here. In fact, both this and barcketsList is only used once. You can substitute one by the other.
Use var when you can, it makes the variable names aligned thus making scanning easier. The variable name itself or its right-hand side assignment should give enough clue usually, plus you can always mouse-over with intellisense.




for (int start = 0; start &lt; symbols.Count(); start++)
{
    char temp = symbols[start];



Since you are just using the index (start) to get the char, you can use a foreach-loop here




if (stack.Count == 0) //only closer no opening
{
    Console.WriteLine("NO");
    flagPrint = true;
    break;



The use of flagPrint here to escape the default final output is a bit more of a hack. For instance, since you are using stack.Any() to check for balance, you can push the closing bracket and break out of the loop to achieve the same behavior. You can also take an another approach and extract a IsBalanced out of existing code and use return.




if (!stack.Any() &amp;&amp; !flagPrint)
{
    Console.WriteLine("YES");
}
else if (!flagPrint)
{
    Console.WriteLine("NO");
}



If we were to keep the flag, this part can be reduced to:

if (!flagPrint)
{
    Console.WriteLine(!stack.Any() ? "YES" : "NO");
}



What is not in the code


IsBalanced is not in a separated method.
Lack of argument checking/error handling


Solution

public void CheckAllExpressionsAreBalanced()
{
    var expressions = new List&lt;string&gt; { "({[]})", "][(]}})(" };
    foreach (var expression in expressions)
    {
        //Console.WriteLine("Expression `{0}` {1} balanced.",
        //    expression,
        //    IsBalanced(expression) ? "is" : "is not");
        Console.WriteLine(IsBalanced(expression) ? "YES" : "NO");
    }
}

// Define other methods and classes here
public bool IsBalanced(string expression)
{
    // argument validation
    if (expression == null)
        throw new ArgumentNullException(nameof(expression));

    // definition
    var brackets = new[] { "()", "{}", "[]" };
    var pairedBrackets = brackets.ToDictionary(x =&gt; x[0], x =&gt; x[1]);

    // non-bracket characters doesn't affect the result,
    // so we can filter them out to make the flow simpler
    Func&lt;char, bool&gt; isBracket = c =&gt; pairedBrackets.ContainsKey(c) || pairedBrackets.ContainsValue(c);

    // implementation
    var stack = new Stack&lt;char&gt;();
    foreach (var c in expression.Where(isBracket))
    {
        var isOpening = pairedBrackets.ContainsKey(c);
        var pair = isOpening
            ? pairedBrackets[c]
            : pairedBrackets.First(x =&gt; x.Value == c).Key;

        if (isOpening)
        {
            stack.Push(c);
        }
        else
        {
            // closing bracket with empty stack
            if (!stack.Any())
                return false;

            // closing bracket doesnt match last opening bracket
            if (stack.Peek() != pair)
                return false;

            // matched
            stack.Pop();
        }
    }

    return !stack.Any();
}


EDIT: I skipped the requirements...
EDIT2: Updated IsBalanced to accept non-bracket characters input. As @HenrikHansen points out, it doesn't make sense to throw an exception.
EDIT3: Fixed issue with trailing brackets.