First, I'll go over the number of permutations. This means that not only do the numbers matter but also their order on the card. There are permut(15,5) = 15!/(15-5)! = 15*14*13*12*11 = 360,360 possible permutations for the B, I, G, and O columns. For the N column, the number of permutations is permut(14,4) = 15!/(15-4)! = 15*14*13*12 = 32,760. Thus, the total number of permutations of bingo cards is 360,3604 × 32,760 = 552446474061128648601600000.
Second, I'll go over the number of combinations. This means that the numbers matter but not their order on the card. There are combin(15,5) = 15!/(5!*(15-5)!) = (15*14*13*12*11)/(1*2*3*4*5) = 3,003 possible combinations for the B, I, G, and O columns. For the N column, the number of permutations is combin(14,4) = 15!/(4!*(15-4)!) = (15*14*13*12)/(1*2*3*4) = 1,365. Thus, the total number of permutations of bingo cards is 3,0034 × 1,365 = 111007923832370565.
On the show, Sheldon asks himself how may UNIQUE bingo cards exist. Based on the later incorrect formulas, I assume he means permutations. In other words, two cards with the same numbers but in different positions would both be unique.
The image above shows Sheldon's formula for the B, I, G, and O columns. He initially gets the formula right at 5! × combin(15,5). However, he incorrectly simplifies it to 15!/(15!-5)!. The second exclamation point shouldn't be there. It should read 15!/(15-10)!. However, he then gets back to the correct answer at 360,360.
We have exactly the same problem with the N column. The formula should be 15!/(15-4)!, not 15!/(15!-4)!. The second exclamation point ruins it.
The ironic thing is later in the Episode, Sheldon become obsesses with errors in the chronology of the Lord of the Rings, much as I'm obsessing with this.