How many bingo card combinations are there? Bingo cards have $5$ rows and $5$ columns. The column names are B,I,N,G,O. Each column has $15$ possible numbers to choose from $1-15$ for $B$, $16-30$ for $I$, $31-45$ for $N$, and so forth.

The third space on the $N$ is 'daubed' or blocked out, so the card only has $24$ numbers.

I don't think $75$ choose $24$ is right. Is $15$ choose $5$ times $4$ plus $15$ choose $4$ (because of the $N$ column) the correct answer$?$ (Which I calculate to $13377$ combinations, which seems too small.)
There are four complete columns of 5 and one with only 4 numbers. For column 1 there are: $$5! {15 \choose 5}$$ Since there are 5 numbers out of a possible 15 used in the column and these can be permuted in any order (in 5! ways). This also holds for the other 3 complete columns. The same method works for the column of 4 giving instead: $$4! {15 \choose 4} $$

Therefore the total answer is: $$ \left (5! {15 \choose 5} \right)^4 4! {15 \choose 4} = 552446474061128648601600000 $$
Since no number is ever repeated on a bingo card under the 'B' the first position has 15 possibilities the second position has 14 the third position has 13 and so on so the correct equation would be 15 * 14 * 13 * 12 * 11 * 4 because B, I, G and O each have that number of possibilities (or 360,360) and under the N where there is a free space it would be 15 * 14 * 13 * 12 (or  32760) 
(360.360 × 4) + 32,760 = 1,474,200 possibilities.