This problem can be approached by looking at the number of distinct ways to assign numbers to the four sides of each card, such that:

1. The sum of the numbers on each card equals 9.
2. No two cards have the same numbers in the same arrangement (i.e., different permutations are allowed).

Step 1: Determine the combinations of numbers on the four sides

The sum of the four numbers on each card is fixed at 9. This means we need to find all possible ways to distribute 9 among four numbers. These numbers can be any non-negative integers, so we're looking for the partitions of 9 into exactly four parts.

Mathematically, this is equivalent to solving the equation:

$x_1 + x_2 + x_3 + x_4 = 9$

where $x_1, x_2, x_3, x_4$ are non-negative integers. The number of solutions to this equation is given by the stars and bars formula:

$\binom{9 + 4 - 1}{4 - 1} = \binom{12}{3} = 220$

So, there are 220 ways to assign numbers to the four sides without considering the arrangement of the numbers yet.

Step 2: Account for the arrangement (permutations)

Each card can have the numbers arranged in different ways (i.e., permutations of the four numbers). Since no two cards can have the same arrangement, we need to count the distinct permutations of each combination.

Since the cards are distinguishable by their arrangement, each of the 220 combinations can be arranged in $4!$ (24) ways if all the numbers on the card are different. However, if there are repeated numbers, the number of distinct arrangements will be less. For example, if two numbers are the same, the number of distinct permutations will be fewer.

Step 3: Consider the total number of distinct cards

Given the problem’s conditions (no repeated arrangement on the same deck), it means the total number of distinct cards will be based on the total combinations of the four numbers, which are distinct in terms of arrangement but not necessarily identical. This directly leads to 220 distinct cards for the deck.

Conclusion:

The deck contains 220 distinct cards.

Would you like further clarification or to explore another aspect of the problem?

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Here are some related questions you might find interesting:

1. How do we compute the number of distinct permutations when there are repeated numbers?
2. Can this problem be generalized to other sums or numbers of sides?
3. How does the stars and bars method apply in other combinatorics problems?
4. What other card game designs use similar combinatoric constraints?
5. How would the deck change if there were more sides on each card?

Tip: The stars and bars method is powerful for problems that involve distributing a fixed sum among several parts (like partitioning 9 into 4 parts).