(April 24, 1997, *Gazette*)

*Have a burning question pertaining to science, engineering, medicine, the arts, music,
humanities, social sciences, physical education, or any other field? Let us know; we'll do our
best to find the answers for you, drawing upon the expertise of members of the university
community.*

Bob French, who has been working in the Utilities Annex of Facilities Management for 22 years, sent an e-mail inqiry to MUNSolved Mysteries. He wrote, "I've always wondered how many different bingo cards can be produced knowing that there are limits of certain numbers under certain letters. Can you tell me the total number of cards, and the formula?" Turns out when the power engineer was going to trade school about 25 years ago, "I asked a physics teacher. He had a slide rule and filled up a chalkboard with numbers, and then we ran out of time. He was probably on the right track, but he never did get the answer worked out. I've wondered off and on ever since," says Mr. French.

Now he can relax. The *Gazette* staff tracked down Dr. Rolf Rees of Mathematics and Statistics.
Dr. Rees is an associate professor specializing in combinatorics, which he describes as "the
science of counting." Here's his answer.

There are C(15,5) = 15 x 14 x 13 x 12 x 11/5 x 4 x 3 x 2 x 1 = 3,003 ways to choose the five numbers (between 1 and 15) to go under the B; similarly there are 3,003 ways to choose each of the five numbers to go under the I, G and O. There are C(15,4) = 15 x 14 x 13 x 12/4 x 3 x 2 x 1 = 1,365 ways to choose the four numbers (between 31 and 45) to go under the N. This gives you a total of 3003 x 3003 x 1365 x 3003 x 3003 (yes, those are multiplications!) ways to choose the 24 numbers which will appear on the card.

Now, I don't know whether order of numbers in a particular column on the card makes a difference, e.g. is B 1 2 3 4 5 different from B 5 4 3 2 1?

If not then the number above gives the total number of cards. If order does make a difference, i.e. if the cards above are considered to be different, then the number of cards possible becomes 3003 x 120 x 3003 x 120 x 1365 x 24 x 3003 x 120 x 3003 x 120. (Given five distinct numbers there are 5! = 120 different ways to order them, and similarly given four distinct numbers there are 4! = 24 ways to order them.)

Thus there are on the order of 100,000,000,000,000,000 bingo cards if the order on columns does not matter; this swells to 500,000,000,000,000,000,000,000,000 if the order on columns does matter. That's a lot of bingo cards.

In a later interview, Dr. Rees added, "I would doubt very much if you could buy a whole set of cards anywhere!"

*There must be zillions of puzzling questions out there! Send yours to MUNsolved Mysteries,*
Gazette, *Arts and Administration Building, Memorial University of Newfoundland, St. John's,
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name and telephone number.*