If you have an all stock portfolio, you should balance several factors to determine how many individual stocks you should have in that portfolio. These stocks should ideally be in several different sectors, or else you'll have concentration and not diversification.
1. If you have too few stocks (4 or less), you will not get enough diversification. If a single stock gets hit hard, your entire portfolio will get hit very hard. By having at least 5 stocks in your portfolio, you get better diversification. Over the long term, studies have shown that a diversified stock portfolio gets a good return with less risk.
2. If you have too much diversification, you may have difficulty keeping track of all your stocks. Owning individual stocks means you have to keep track of the situations on all your stocks. Can you manage a stock portfolio of 20?
3. If the total amount in your portfolio is not that big, you should have less positions. Otherwise, commissions will be too big a percentage to position size. Let's say it costs $10 to buy and sell a stock, and each position of a stock in the portfolio is $500. That means you are spending $20 per trade, or $20/$500 = 4%. For every investment or trade you make, you are spotting the market 4%! That's too much.
4. If the total amount in your portfolio is much bigger, you can diversify more to reduce risk.
5. If you are very young (college, early 20s), you can take a lot more risk in your portfolio. Even if you lose the money, you have time to make it up. So you can be more aggressive, and you could even buy two stocks at $1500 a position. Of course, this money should be discretionary money, and not retirement money, or money that you need.
So, to summarize, in general, Jim Cramer in his book "Real Money: Sane Investing in an Insane World", suggests having a portfolio between 5-10 individual stocks. This is enough for diversification, but not too much that it becomes too much to handle.
If you have the time and inclination, you can hold up to 20 individual stocks, but only if you can handle it.
Alternatively, you can use ETF (Exchange Traded Funds) to supplement your portfolio. You can invest in 5 individual stock for 50% of the portfolio, and in the other 50%, invest in broadbased ETFs such as EEM (Emerging International Market ETF) or EFA (Developed International ETF). This way, you can explore with 50% of your portfolio by investing in individual stocks, and the other 50% for a core ETF position or ETFs that have exposure that your individual stocks lack. For example, if 5 of your individual stocks are all US Domestic stocks, you can use the other 50% to invest in International ETFs as mentioned above.
If you don't have the time or inclination
If you don't have the time and inclination to study and research individual stocks, maybe you should consider a diversified portfolio of ETFs or Mutual Funds instead.