Number of possible 75-Ball Bingo Cards

Last Up date: 2018 December 14 on nuMacMini work/gd/lafarr/math
Recent changes: fixed "O" goes from 62 to 75. New heading after about 4 paragraphs
This Page started on: 2008 January 27 11:51:50

There are 5 Columns on a Bingo Card, each labeled with the letters from BINGO. Under each letter can be 5 of 15 possible numbers, and no number appears more than once on any Bingo Card. B's go from 1 through 15, I's go from 16 through 30, N's go from 31 through 45, G's go from 46 through 60, and O's go 61 through 75.

The number of ways X items can be select from M things is:
X for the first one
(X-1) for the second
(X-2) for the third
and so on. Multiplying these all together is the number of ways!

A simple example would be: How many two letter "words", with no repeating letters, can be formed from the letters "ABCDE"? The first could be any one of the 5 letters and the second could be any of the remaining 4. So the number of ways would be 5 times 4 or 20. This would also be the number of ways a President and Vice-President could be selected from a group of five people.

There would be 60 possible three letter words.

A word of warning here: In the above examples order counts. "BD" is a different word than "DB", the same for President & Vice President. If we were just selecting a committee of two from five people it would not mater what order they were "drawn" it would result in the same committee. For this: We have to divide by the number of ways two could be selected from the two drawn. For this example it would be 2 times 1 or two. Only 10 committees could be selected from 5 people.

Back to 75 Ball BINGO Card calculations

Order drawn does count. For the "B" column we are selecting 5 things from 15, and if we ignore the "Free" cell in the center it is the same for each of the 5 columns.

15*14*13*12*11 = 360,360

If we multiply 360,360 by itself five times we get:


This would be the number of different BINGO cards if all the columns were the same. No "Free" space under the "N" But if we only allow four values for the N Column, the 11 should not have been multiplied in for the N column. We can take it out by dividing the above number by 11. Giving us:

Possible BINGO cards!
This agrees with the number calculated on a WiKi page.

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