This looks like a homework assignment, and we don't generally do homework for people here. However, that thought hand't occurred to me before I came up with the answer, so consider yourself lucky thata you got a response at all.
FWIW, you should find another isntructor because he's not teaching you the arguably most important programmer skill - analysis.
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Split the string into an collection of words, and then use LINQ to count the number of instances of a given word.
string mywordString = "around the rugged rock the ragged rascal ran";
string findWord = "the";
List<string> words = new List<string>();
words.Add(words.ToLower().Split(" "));
findWord = findWord.ToLower();
int count = (from word in words
where word == findWord
select word).Count();
You may have to tweak it a little, but it provides the gist of how you could approach the problem.