Question 1164826
To show this identity, we will count the number of elements in the set  where .

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### Method 1: Summing based on the value of 

In this method, we fix the value of  and count how many pairs  satisfy the condition  and .

1. The possible values for  are  because  must be strictly greater than  and , and the smallest value in  is 1.
2. If we fix  (where  ranges from  to ):
* The values of  must be chosen from the set . There are  choices for .
* The values of  must also be chosen from the set . There are  choices for .


3. For a fixed , there are  possible pairs .
4. Summing over all possible values of  from  to :



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### Method 2: Counting by Cases of  and 

Now, we count the same set  by considering the relationship between  and . There are two main scenarios: either  or .

**Case 1: **
If , then the condition  and  simplifies to . We just need to choose two distinct numbers from  and assign the larger one to  and the smaller one to  (and ).

* Number of ways to choose 2 distinct elements from : .

**Case 2: **
If , we are choosing three values  such that  is the largest. First, we choose 3 distinct numbers from the  available numbers. Let these be  where .

* To satisfy the condition,  **must** be the largest value ().
* For the remaining two values ( and ), one must be  and the other must be .
* There are 2 possibilities:  or .
* Number of ways to choose 3 elements and arrange them: .

### Conclusion

By combining the two cases in Method 2, we get:


Since both methods count the exact same set , we have proven that:


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Would you like me to show how this expression simplifies further to the standard formula ?