To solve the problem of placing the numbers 1, 2, 3, and 4 in an 8x8 table such that each row has the numbers in ascending order from left to right, we can approach it by recognizing the constraints of the placements.

1. Understanding the Constraints:

   • Each row can have any combination of the numbers 1, 2, 3, and 4.
   • The arrangement must be in non-decreasing order within each row.

2. Using Combinations:

   • Each row can be seen as a combination problem where we want to select positions for the numbers 1, 2, 3, and 4 in a row of 8 cells.
   • The main task is to count how many different ways we can fill each row such that the chosen numbers are in non-decreasing order.

3. Stars and Bars Method:

   • We can think of this as a "stars and bars" problem where we want to place 4 indistinguishable objects (the counts of 1s, 2s, 3s, and 4s) into 8 distinguishable boxes (the cells of a row).

   • Let $x_1, x_2, x_3, x_4$ be the number of 1s, 2s, 3s, and 4s respectively in a row. We need to satisfy: $x_1 + x_2 + x_3 + x_4 = 8$

   • The number of non-negative integer solutions to this equation can be found using the "stars and bars" theorem, which gives us: $\binom{n + k - 1}{k - 1}$ where $n$ is the total number (8) and $k$ is the number of categories (4).

   • Thus, the formula will be: $\binom{8 + 4 - 1}{4 - 1} = \binom{11}{3}$

4. Calculating the Binomial Coefficient:

   • Now, calculating $\binom{11}{3}$: $\binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165$

5. Total Combinations for All Rows:

   • Since each of the 8 rows can be filled independently, the total number of ways to fill the entire table is: $165^8$

Thus, the total number of ways to place the numbers 1, 2, 3, and 4 in the 8x8 table such that each row has the numbers in ascending order is $165^8$.

Summary

The answer is $165^8$.

Would you like further details or have any questions about the steps? Here are some related questions you might consider:

1. How does the "stars and bars" method work in general?
2. Can you provide an example of filling one row of the table?
3. What if we wanted to use different numbers or a different grid size?
4. How would you calculate the arrangement if there were restrictions on certain numbers?
5. What are some practical applications of this type of combinatorial problem?

Tip: When solving combinatorial problems, always look for ways to apply well-known techniques like "stars and bars," which can simplify counting arrangements significantly.