The Coca-Cola Company donated 38 cases of Coke products for the Mars Middle School annual picnic. There are 4 dozen bottles per case. Each bottle contains 16 ounces of beverage.

How many pints of Coke did the Coca-Cola Company donate?

The Coca-Cola Company donated 38 cases of Coke products, which contain 4 dozen bottles of Coke per case. This means that the Coca-Cola Company donated 38 cases * 4 dozen bottles/case = <<38*4=152>>152 dozen bottles of Coke.

Since there are 12 bottles in a dozen, the Coca-Cola Company donated a total of 152 dozen bottles * 12 bottles/dozen = <<152*12=1824>>1824 bottles of Coke.

Each bottle of Coke contains 16 ounces of beverage, so the Coca-Cola Company donated a total of 1824 bottles * 16 ounces/bottle = <<1824*16=29184>>29184 ounces of Coke.

There are 16 ounces in a pint, so the Coca-Cola Company donated a total of 29184 ounces / 16 ounces/pint = <<29184/16=1824>>1824 pints of Coke. Answer: \boxed{1824}.