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.)
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$\begingroup$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: $$ \left (5! {15 \choose 5} \right)^4 4! {15 \choose 4} = 552446474061128648601600000 $$
$\endgroup$ 4 $\begingroup$The answer is ${15\choose5}^4\cdot{15\choose4}$.
Columns B, I, N, G and O are independent, so the Fundamental Counting Principle applies. You multiply the number of choices for each column.
Edit: Oops! @Zestylemonzi is correct. I should have used permutations, not combinations. The answer should be $({}_{15}P_{5})^4\cdot({}_{15}P_{4})$.
$\endgroup$ 0 $\begingroup$It is not Permutations, because the order does not matter. It is in fact Combinations. $$ {15 \choose 5}{15 \choose 5 }{ 15 \choose 4}{15 \choose 5}{15 \choose 5} = 1.11\cdot10^{17} $$
$\endgroup$ 2 $\begingroup$It is a combination in respect to "full-card" bingo, only. However, not every game of Bingo is played as full-card. Other variations I have played include "X", "Crazy T", "Crazy L", "Butterfly", and then straight line/4 corners. With regard to those games, the order of the numbers certainly does matter, so it is a permuation. Therefore, it is (15P5)^4 X 15P4. According to wikipedia's Bingo page, it is 552,446,474,061,128,648,601,600,000. I tried solving this in Excel, and had an answer less accurate as it seems Excel has a max of 15 significant digits. It does seem that wikipedia's answer is accurate. 15P5=360,360 and 15P4=32,760. When you multiply 360,360^4 x 32,760, your answer will end with 5 zeros.
In regards to full-card bingo, where the order does not matter, then we have (15C5)^4 x 15C4, which is 111,007,923,832,370,565.
$\endgroup$ 1 $\begingroup$@Zestlyemonzi has correctly given the number of possible Bingo cards. Note, however, that Bingo cards have one axis of symmetry; if you reverse the order of the numbers in each column, the card will produce the same results in a standard game (not in all variants). Therefore, for a standard game, the number of distinct Bingo cards is 552446474061128648601600000 / 2 = 276223237030564324300800000.
$\endgroup$ 1 $\begingroup$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.
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