For how many pairs of positive integers n and k with n and k less than or equal to 20, is the number (2n)!(2k)!n!k!(n+k)! an integer?
To determine the number of pairs of positive integers (n, k) for which the given expression is an integer, we can break down the problem into smaller parts. Let's go step by step:
1. Consider the number (2n)!:
- We know that the factorial of a positive integer is always an integer.
2. Next, consider the number (2k)!:
- Similar to (2n)!, the factorial of a positive integer is always an integer.
3. The expression n!:
- Again, the factorial of a positive integer is always an integer.
4. The expression k!:
- The factorial of a positive integer is always an integer.
5. The expression (n+k)!:
- The factorial of a positive integer is always an integer.
From the above analysis, we can see that all the individual components of the given expression are integers.
To determine the total number of pairs (n, k) for which the expression (2n)!(2k)!n!k!(n+k)! is an integer, we can consider all possible values of n and k less than or equal to 20 and count the valid pairs.
Since both n and k are positive integers less than or equal to 20, there are a total of 20 possible values for each. Therefore, we need to check the expression for all possible pairs of n and k.
Calculating the expression (2n)!(2k)!n!k!(n+k)! for all pairs (n, k) with n and k less than or equal to 20 would be time-consuming. Instead, we can use a computer program or a mathematical software/tool to compute the expression for all 400 (20*20) pairs and count how many yield an integer value.
Using programming languages like Python, we can write a code snippet to perform this calculation:
```python
count = 0
for n in range(1, 21):
for k in range(1, 21):
expr = ((2*n) * (2*k) * n * k * (n+k))
if expr % 1 == 0:
count += 1
print(count)
```
When you run this code, it will calculate the number of pairs (n, k) for which the given expression is an integer.