Bingo Blitz - Card Possibility

This is one for the nerds. Do you know how many different cards there can be in a standard 75-ball bingo game where the middle square is covered? Read on to find out!

Bingo Blitz follows a standard 75-ball bingo game where the middle square of each card is covered already. Each square is 5x5, meaning each card has 25 squares. With one already covered, this leaves 24 numbers needed for the rest of the card. It is not as simple as choosing 24 numbers from 75 because he bingo cards are split into 5 columns, namely B, I, N, G and O. Combinations and permutations are taken from each of the columns and put together to make a full card.

This is just one of many possible bingo card combinations

Column B uses numbers 1-15
Column I uses numbers 16-30
Column N uses numbers 31-45
Column G uses numbers 46-60
Column O uses numbers 61-75







The number of possible bingo cards here can be calculated by using some mathematics i.e. permuations. This is simply how many different ways are there picking a certain number of items from the whole pool of items where the order of the items picked is important. As stated, this uses 5 different columns and so the permutation for each column would need to be worked out. This is done using this formula:

Permutation formula

Where P is for permutation
            n is for number of items to chose from
            r is number of items to be chosen

The exclamation mark represents factorial. It means multiplying all of the numbers together up to and including the number before the exclamation mark. For example,

2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! 4 x 3 x 2 x 1 = 24

and so on.


In this case, n is 15. Columns B, I, G and O have r as 5, where as column N has r as 4 because only 4 numbers need to be chose for this column.

So, as can be seen, columns B, I, G and O all have the same formula. Essentially, we place n = 15 and r = 5 into the formula above. This is:


15! divided by 10!

There will be some cancellations in this here as both 15! and 10! contain the first 10 numbers. The permutation for columns B, I, G and O is,

 11 x 12 x 13 x 14 x 15 = 360360

Now, the permutation for column N is n = 15 and r = 4. This is

15! divided by 11!

Again, there is some cancellation here because both contain the first 11 numbers. This leaves:

12 x 13 x 14 x 15 = 32760

Now is where we put all of these together using multiplication because of how many permutations there are in having just 2 columns different. We simply multiply the 2 numbers in bold from above, but keeping in mind that 360360 is to be used 4 times for the 4 columns of B, I, G and O.

Our answer is:

360360 x 360360 x 32760 x 360360 x 360360 = 552,446,474,061,128,648,601,600,000

So there are 552,446,474,061,128,648,601,600,000 different possible cards! It is safe to say that no one will have the same card as you.