How Many Possible Unique 75-Ball Bingo Cards Are There?

75 ball bingo ticketShould you be thinking it is possible to cheat at 75-Ball bingo by printing out all the possible cards before a game you may want to think again.  The thing is despite being a relatively simple game with an easy to understand layout, that can only feature 75 possible balls, the actually number of unique cards you can produce is astronomical.

We are not talking thousands, millions, billions or even trillions of possible cards, there are in fact 552 septillion possible card combinations for 75-ball bingo.  If you write that number out it looks like this with 552 followed by a further 24 digits:

552,446,474,061,128,648,601,600,000 or 552 Septillion possible unique 75-ball bingo cards.

If you say that number out loud it would be five hundred fifty two septillion, four hundred forty six sextillion, four hundred seventy four quintillion, sixty one quadrillion, one hundred twenty eight trillion, six hundred forty eight billion, six hundred one million, six hundred thousand.

To put this into some context if we could print one card each second it would allow us to product 31,536,600 cards each year.  Yet it would take 17,517,629,486,410,350,152 years of constant printing to achieve that.  That is 17.5 quintillion years, the universe itself has only been around for 13.8 billion years, meaning it would take (roughly) 1.25 billion times the age of the universe to complete that printing.  Safe to say then you can’t use this method to cheat at 75-ball bingo.

75 Ball Bingo Tickets

The numbers here are so big and incomprehensible that they even blow the possible number of 90-ball bingo cards out of the water.  In 90-ball bingo there are 24 quadrillion possible unique cards, which is still staggeringly high, but 75-ball bingo has over 23 billion times that in unique combinations, despite having less numbers overall in the game.

Let’s say you wanted to print all of those cards out and for efficiency you print them small and fit 100 cards onto a single sheet of paper.  Trees can produce, on average, around 10,000 pieces of paper and there are around 3 trillion trees on the earth right now.  It would take 184 million planets worth of trees to produce that many cards.  When we say astronomical numbers we really meant it.

If you want to delve into how we did this and the logic behind it then we show you how we got to this incredible number below.

75-Ball Bingo Card Combinations

How many bingo cards are there?

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

Each card has 24 numbers, split into 5 columns which each select from a fifth of the possible numbers (so e.g. if the number 63 is present on a card, it will always be in the fifth 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 75-ball bingo card contains 5 numbers (except for the third/central column, which contains only 4 numbers). These numbers are selected from 15 possibilities: for example, the first column can contain any 5 numbers from 1 to 15 inclusive.

On a 75-ball bingo card, the order in which the numbers appear is significant and can vary. For example, a card with the numbers 4, 11, 2, 15, 7 in the first column is different from one with the same numbers in the order 11, 15, 7, 2, 4 – because you are looking to match rows, columns or (in some games) other patterns on your card, these two differing orderings of numbers in any column will give you different results.

All of this means that we are not just interested in the number of combinations of 5 numbers from 15, but the number of permutations.

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

75 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 = 15 and r = 5), we are able to calculate that there are 360,360 permutations for a standard column.

For the third column, only 4 numbers are selected so by substituting r = 4 into the formula instead, we find there are only 32,760 permutations for this column.

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

For each possible arrangement of the first column, the card could 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 of the third column, and so on.

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

360,360 x 360,360 x 32,760 x 360,360 x 360,360 = 552,446,474,061,128,648,601,600,000

Over five-hundred and fifty-two septillion!

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 only 4 numbers in a few of the columns.

from math import perm, prod
from typing import Tuple

def total_possible_cards(
    numbers_by_column: Tuple[int, ...],
    possibilities_by_column: Tuple[int, ...]
) -> int:
    Calculate the total number of different possible bingo cards.
    return prod(
        perm(possibilities_by_column[col], numbers)
        for col, numbers in enumerate(numbers_by_column)

    numbers_by_column=(5, 5, 4, 5, 5),
    possibilities_by_column=(15, 15, 15, 15, 15)