# Number of distinct UK bingo pages

An UK bingo ticket contains 27 spaces, arranged in nine columns by three rows. Each row contains five numbers and four blank spaces. Each column contains up to three numbers, which are arranged as follows:

• The first column contains numbers from 1 to 9,
• The second column numbers from 10 to 19,
• The third column number from 20 to 29 so on up until the last column, which contains numbers from 80 to 90.

Tickets are created as strips of 6, in such a way that every number from 1 to 90 appears exactly once. This is also called a bingo page. (Wikipedia page)

Example of a valid bingo page: $$\begin{array}{ | c |} \hline \begin{array}{ | c | c | c | c | c | c | c | c | c | } \hline 3 & 12 & & 33 & & & & 70 & 83 \\ \hline 6 & 16 & 20 & & & & 61 & 75 & \\ \hline & & 29 & & 43 & 56 & 65 & 78 & \\ \hline \hline \end{array} \\ \begin{array}{ | c | c | c | c | c | c | c | c | c | } \hline & 11 & 26 & 30 & 40 & & 60 & & \\ \hline 7 & 17 & & & 46 & 58 & & & 88 \\ \hline & & & 32 & 49 & & 69 & 77 & 89 \\ \hline \hline \end{array} \\ \begin{array}{ | c | c | c | c | c | c | c | c | c | } \hline & 10 & 23 & 34 & & 54 & & & 90 \\ \hline 1 & & 25 & & & 57 & 66 & 73 & \\ \hline & 19 & & 38 & 48 & 59 & 68 & & \\ \hline \hline \end{array} \\ \begin{array}{ | c | c | c | c | c | c | c | c | c | } \hline & 13 & 28 & & & 50 & 62 & 74 & \\ \hline 4 & & & 37 & 45 & 55 & & & 80 \\ \hline 5 & & & 39 & & & 64 & 79 & 84 \\ \hline \hline \end{array} \\ \begin{array}{ | c | c | c | c | c | c | c | c | c | } \hline & 15 & 21 & & 44 & 52 & & & 81 \\ \hline 2 & & 22 & 35 & & & 67 & & 82 \\ \hline 9 & & & 36 & & 53 & & 71 & 86 \\ \hline \hline \end{array} \\ \begin{array}{ | c | c | c | c | c | c | c | c | c | } \hline & 14 & 24 & 31 & 41 & & & 72 & \\ \hline 8 & & 27 & & 42 & & & 76 & 85 \\ \hline & 18 & & & 47 & 51 & 63 & & 87 \\ \hline \hline \end{array} \end{array}$$

How can one calculate how many different bingo pages can be created?

• In a single ticket (three rows by nine columns), can a column be empty, or must a column contain one to three numbers? – awkward Nov 9 '16 at 15:04
• @awkward: There are no constraints on the number of numbers in a column in a ticket. – boozo Nov 12 '16 at 15:31