How Many Possible Unique 50-Ball Bingo Cards Are There?

50 Ball Bingo50-Ball bingo is like 90-ball bingo lite, it plays out in a very similar way but with less balls, making the games quicker.  It is kind of like an intermediate between 90-ball and 30-ball (speed bingo).

Cards are made up of just two columns with 5 numbers each (10 numbers total) with no blanks to deal with.  Still, in spite of this easy format you might be surprised to hear that the number of possible unique cards you can make is still massive.  The number comes out at just under 6 billion (5,904,900,000).

Should you be in any way worried that someone could cheat at 50-ball bingo by bringing all the possible card combinations in advance with them then now you can see this isn’t practical in any way.  It is also not possible to pre-buy all the possible cards for a game either given the shear number of possibilities.

50 Ball Bingo Tickets

To give some context to these numbers if you wanted to print all the possible cards and you are using a  printer that prints one card per second then you can print 31,536,600 cards per year.  It would, therefore, take you over 187 years to print all of those cards out.  Let’s say you wanted to buy all the tickets for a game at 5 pence each, this would cost you a staggering £295 million to do so.

This is no where near the number of possible bingo cards for a game like 90-ball or 75-ball but it is still sufficiently high to ensure that no one could ever possess all the possible combinations for a given game.  The number is still, however, 3417 times bigger than the number of possible cards for the smaller 30-ball game.

To see how we got to this number an the logic behind it read on below.

50-Ball Bingo Card Combinations

50-ball bingo code stats example
Example 50-Ball Bingo Card

How many bingo cards are there?

That’s an interesting question, but a solvable one, with a bit of mathematics!

Each card has 10 numbers, split into 5 columns which each select from a fifth of the possible numbers (so e.g. if number 25 is present on a card, it will always be in the third column).

This means that each column in the card is effectively independent of the others: two cards could have identical first columns, but be different in the other columns. This means we can calculate the number of different possible cards by calculating the different possibilities for each column, and multiplying the results together.

Part 1: How many possibilities are there for one column?

Each column of a 50-ball bingo card contains 2 numbers. These numbers are selected from 10 possibilities: for example, the first column can contain any 2 numbers from 1 to 10 inclusive.

On a 50-ball bingo card, the order in which the numbers appear is significant and can vary. For example, a card with the numbers 15 at the top of the second column and 17 below it is different from one with the same numbers in the opposite order – because one method of winning is to complete a row, these two different orderings of numbers in any column will give different results.

All of this means we are not just interested in the number of combinations of 2 numbers from 10, but the number of permutations.

Using the standard mathematical formula for the number of permutations without replacement…

50 ball bingo card calculation formula

(where n is the number of possibilities and r is the number of selections we are making, i.e. in our case n = 10 and r = 2), we are able to calculate that there are 90 permutations for each column.

Part 2: How many possibilities are there for the entire card?

For each possible arrangement of the first column, the card contain any of the possible arrangements for the second column. And then for each of those possibilities, the card could contain any of the possible arrangements for third column, and so on.

Therefore, to calculate the total number of possible cards, we multiply together the possibilities for each column:

90 x 90 x 90 x 90 x 90 = 5,904,900,000

Over 5 billion!

Python Code Used For The Calculations

Below is the Python code for calculating this – so if you’re planning or playing a slight variety of the game, you could tweak this and see how the total number of cards is affected if e.g. there are 12 possible numbers in each column instead of 10.

from math import perm, prod

def total_possible_cards(columns: int, numbers_per_column: int, possibilities_per_column: int):
    """
    Calculate the total number of different possible bingo cards.
    """
    return prod(
        perm(possibilities_per_column, numbers_per_column)
        for _ in range(columns)
    )

total_possible_cards(5, 2, 10)
5904900000