Big O Notation

Leaving aside questions of efficiency, the main problem here is dealing with punctuation and trying to decide what constitutes a 'word'.

I've ignored capitalization and apostrophes but treated a hyphenated (or underscore separated) word as being a single word. I've also treated '&', surrounded by spaces, as a separate word and all numbers including embedded decimal points and thousand separators as being 'words'.

In the belief that a conventional approach would probably  be more efficient here than either regular expressions or LINQ, I've used the former. You'll see I've sorted the array of words to make it easier to count duplicates, though this might be less efficient than other approachs depending on the number of words in the paragraph.

I've just assumed that the file contains a single paragraph rather than trying to parse out the first paragraph or anything like that:

using System;
using System.IO;
using System.Collections.Generic;

class Program
{

     static void Main()
     {
         string text = File.ReadAllText("jan.txt"); // or whatever

         Console.Clear();
         Console.WriteLine("The paragraph is :\n");
         Console.WriteLine(text);
         Console.WriteLine();

         // convert everything to lower case
         text = text.ToLower();
         char[] chars = text.ToCharArray();

         // replace most non-alphanumeric characters with spaces
         for (int i = 0; i < chars.Length;i++)
         {
            if (!Char.IsLetterOrDigit(chars[i]))
            {
               if (chars[i] == ' ') // space
               {
                  continue;
               } 
               else if (chars[i] == '\'') // apostrophe
               {
                  chars[i] = ' ';
                  if (chars.Length - 1 > i && chars[i+1] == 's')
                  {
                     chars[i+1] = ' ';
                     i++;
                  } 
               }
               else if (chars[i] == '-' || chars[i] == '_') // hyphen or underscore
               {
                  if ((i > 0 && i < chars.Length - 1) && (Char.IsLetter(chars[i - 1]) && Char.IsLetter(chars[i + 1])))
                  {
                     continue;
                  }
                  chars[i] = ' ';
               }
               else if (chars[i] == '&') //ampersand
               {
                  if ((i > 0 && i < chars.Length - 1) && (chars[i - 1] == ' ' && chars[i + 1] == ' '))
                  {
                     continue;
                  }
                  chars[i] = ' ';
               }
               else if (chars[i] == '.' || chars[i] == ',') // dot or comma
               {
                  if ((i > 0 && i < chars.Length - 1) && (Char.IsDigit(chars[i - 1]) && Char.IsDigit(chars[i + 1])))
                  {
                     continue;
                  }
                  chars[i] = ' ';
               }     
               else
               {
                  chars[i] = ' ';
               }
            } 
         }
 
         text = new string (chars);
 
         // get all words
         string[] words = text.Split(new char[]{' '}, StringSplitOptions.RemoveEmptyEntries);

         // sort into alphabetic order so indentical words are adjacent to each other
         Array.Sort(words);

         // obtain list of unique words and their frequencies
         List uniqueWords = new List();
         List wordCounts = new List();

         uniqueWords.Add(words[0]);
         string currentWord = words[0];
         int count = 1;

         for (int i = 1; i < words.Length; i++)
         {
            if (currentWord == words[i])
            {
               count++;
            }
            else
            {
               wordCounts.Add(count);
               uniqueWords.Add(words[i]);
               currentWord = words[i];
               count  = 1;
            }
           
            if (i == words.Length - 1)
            {
               wordCounts.Add(count);
            }
         }

         // display results
         Console.WriteLine("The word analysis is:\n");
         Console.WriteLine("Number of unique words : {0}\n", uniqueWords.Count);   

         for (int i = 0; i < uniqueWords.Count; i++)
         {
            Console.WriteLine("{0} : {1}", uniqueWords[i], wordCounts[i]);
         }

         Console.ReadKey();                    
     }
}