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: $$ \left (5! {15 \choose 5} \right)^4 4! {15 \choose 4} = 552446474061128648601600000 $$