In BINGO, a 5x5 card is filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares.

A card is made by placing:
- 5 numbers from the set 1-15 in the first column,
- 5 numbers from 16-30 in the second column,
- 4 numbers from 31-45 in the third column (skipping the WILD square in the middle),
- 5 numbers from 46-60 in the fourth column,
- 5 numbers from 61-75 in the last column.

To play BINGO, someone names numbers, chosen at random, and players mark those numbers on their cards. A player wins when they mark 5 in a row, either horizontally, vertically, or diagonally.

How many distinct possibilities are there for the values in the diagonal going from the top left to the bottom right of a BINGO card, in order?

Example BINGO card:

\[
\begin{array}{ccccc}
5 & 16 & 35 & 46 & 75 \\
4 & 17 & 34 & 47 & 74 \\
3 & 18 & \text{Wild} & 48 & 73 \\
2 & 19 & 32 & 49 & 72 \\
1 & 20 & 31 & 50 & 71 \\
\end{array}
\]

To find the distinct possibilities for the values in the diagonal going from the top left to the bottom right of a BINGO card, we need to consider the ranges of numbers that can appear in each column.

The first column can have any 5 numbers from the set 1-15. There are 15 numbers in this range, so there are "15 choose 5" possibilities for the numbers in the first column.

The second column can have any 5 numbers from the set 16-30. Again, there are 15 numbers in this range, so there are "15 choose 5" possibilities for the numbers in the second column.

The third column has a Wild square in the middle, so we need to skip it and consider the remaining 4 squares. The numbers in the third column can come from the set 31-45, which has 15 numbers. Therefore, there are "15 choose 4" possibilities for the numbers in the third column.

The fourth column can have any 5 numbers from the set 46-60, which has 15 numbers. So there are "15 choose 5" possibilities for the numbers in the fourth column.

The last column can have any 5 numbers from the set 61-75, which again has 15 numbers. So there are "15 choose 5" possibilities for the numbers in the last column.

To find the total number of distinct possibilities for the diagonal, we multiply the number of possibilities for each column together:

"15 choose 5" "15 choose 5" "15 choose 4" "15 choose 5" "15 choose 5".

Evaluating this expression, we find:

(3003) (3003) (1365) (3003) (3003) = 13,601,464,112,541,695.

Therefore, there are 13,601,464,112,541,695 distinct possibilities for the values in the diagonal going from the top left to the bottom right of a BINGO card, in order.

Learn more about Possibilities here:

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