Rashi - The Sapirstein Edition
TWENTY-TWO THOUSAND.  But broken down into their components, you find three hundred more, as follows:  The children of Gershon, seven thousand, five hundred (8), the children of Kohat, eight thousand, six hundred(9); the children of Merari, six thousand, two hundred(10).  Thus, there is a total of twenty-two thousand, three hundred. 
Why did [Scripture] not include [the other three hundred] with the rest of the Levi'im, and let them redeem, i.e., be the redemption of the firstborn, and then the two hundred and seventy-three firstborn who were in excess over the number of Levi'im would not have required redemption through money?(1)   
Our Rabbis have said in Tractate Bechorot(2) those three hundred Levi'im not included in the total were themselves firstborn, and it was enough for them that they should exempt themselves from the redemption. (3)
8. vs. 22
9. vs. 28
10. vs. 34
1. vs. 46 below
2. Bechorot 5a; see also Bamidbar Rabbah 3:14
3. The twenty-two thousand Levi'im mentioned by our verse were those who redeemed twenty-two thousand of the firstborn.  The other three hundred Levi'im were themselves firstborn.  Their Levitic status served as redemption for their own sanctity as firstborn.  They could thus not serve as redemption for others.