# Arrangements of bingo cards with constraints

Bingo Players Club adopted new rules in bingo.

Letter B covers 1-15, and the arrangement of numbers shall always be ascending;

Letter I covers 16-30, and the arrangement of numbers shall always be descending;

Letter N covers 31-45, and the arrangement of numbers shall be random, such that no two prime numbers are consecutive;

Letter G covers 46-60, and the interval of each number is two; in increasing order; and

Letter O covers 61-75, and the interval of each number is three; in decreasing order.

Furthermore, for the row arrangement, it should be B is less than I, I is less than N, N is less than G, G is less than O.

How many pieces of bingo cards can be produced by the company if in the said rule, they will adapt a new card with 5x4 boxes, 5 rows and 4 columns, with NO free slot?

• I assume you mean a 5x5 board? If not, which letter is left out? Try thinking with combinations and multiplication principle. – JMoravitz Nov 29 '14 at 14:04
• Sorry, I'll edit it. I mean its a 5x4 card. – JP Punzalan Nov 29 '14 at 14:39
• I still think you are confused. Bingo cards are almost always constructed with 5 columns, each column corresponding to one of the letters in the word BINGO. A 4x5 card would make sense, but 5x5 cards are the most common. – JMoravitz Nov 29 '14 at 15:54

I expect that your problem statement is still slightly misworded. The question is likely meant to ask "how many different pieces of bingo cards can be produced." Bingo cards generally have a column for each letter in the word BINGO, so generally exactly 5 columns. An example bingo card using the rules defined above would be something like this:

$\matrix{ & B & I & N & G & O\\ & 1 & 29 & 32 & 48 & 74\\ & 3 & 26 & 36 & 50 & 71\\ & 5 & 23 & 33 & 52 & 68\\ & 6 & 22 & 31 & 54 & 65\\ & 9 & 18 & 39 & 56 & 62}$

A good way to go about thinking of this problem would be to break it up into steps to use the multiplication principle. Find how many ways there are to construct the first column with the given rules, find how many ways there are to construct the second column with the given rules, $\dots$, find how many ways to construct the fifth column with the given rules. The total number of different cards would then be the product of the number of ways of creating each column.

Hints for dealing with each column:

For columns 1 and 2: note that if you have picked five numbers, there is only one way they could be arranged since it says they must be in ascending(/descending) order. As such there is a bijection between finding the number of ways of setting up the column to the number of ways of choosing a combination of 5 numbers from the available numbers to that column.

For column 3: Figure out which numbers are prime in that range. To check if a number is prime you can either reference a list of primes, or it suffices to check if a smaller prime less than or equal to the number's squareroot divides evenly into it (in this case, check if any of the primes 2,3,5, or 7 divide evenly into the number). Once you know what the primes are, you can use an inclusion/exclusion principle argument to find the number of arrangements which avoid two adjacent primes.

For columns 4 and 5, if I'm understanding correctly, the interval is exactly 2( or 3). In which case, the only free variable is from what number did it start so that it still fits within the available range. In my example above, for column $G$, I chose to begin with 48. I could have started with 49 instead, in which case it would have been 49,51,53,55,57. What is the smallest number I could have started with? What is the biggest?

Again, once you have figured out number of possibilities of each column, the total number of possibilities is:

$$\#(total) = \#(col1)\cdot\#(col2)\cdot\#(col3)\cdot\#(col4)\cdot\#(col5)$$

If it turns out that there are meant to be four rows, the same logic applies, just adjust the specifics of the calculations to accommodate that. If there are indeed meant to be 4 columns as written, then you need to find out what exactly is meant by a bingo card with 4 columns but 5 letters as they are not in any way common.