How many stocks to duplicate an index??? --

OK for you academics. I know the answer is in my old college statistics book but after 45 years I'm having a hard time finding the book. You are collectively the smartest bunch of investors I know So I'm throwing this one to all of you.

I recently saw different TV interviews and articles quoting Jack Bogle, former CEO of Vanguard and inventor of the low cost index fund and Burton Malkiek, professor and author of Random Walk Down Wall Street. Both think that since 85% of all professional money managers, after fees and expenses fail to beat the indexes that investors should just buy a low fee index fund and let it ride.

1 = Suppose you could have a NO expense index fund. Given the my and my wife's IRA give us both 100 free trades a year and even after we eat up the 100 free trades we get trades for $8.95 per trade, so since my average traded is about $100,000 per trade that is still only a .00895% commission; our accounts are really close to no cost.

2 - Suppose I wanted to duplicate the S&P 500 Index. How many random S&P 500 stocks would I have to buy till my tracking error was almost nil??? 25, 50, 100, 38 ???

3- Suppose that instead of buying a cap weighted amount I just bought an equal $ amount of each stock. Since smaller stocks now have an equal weight does that give me an advantage to beat a cap weighted index?

4 - A proof that smaller stocks give a higher rate of return over time and that arithmetic indexes give a higher rate of return over Cap Weighted index is a chart below: Over the last 15 years S&P 500 large cap = 35.30%, S&P 400 mid cap = 174.66% S&P 600 small cap = 216.45% and the Value Line Index of 1700 non weighted stocks returned 308.00%.

So, do you think that if I took 1700 stocks of the Value Line Index that appear to overlap the S&P 500, 400 and 600 Indexes and bought an equal value of what ever the number is of stocks needed to duplicate the index I could actually not only track the index exactly, thus beating even those funds that only charge .35% fees but actually beat the S&P 500 index by duplicating the Value Line Index with no cost or fees???

I hope I've made my assumptions clear

Have at it