Introduction:
In this problem, we are given a language with 6 letters and we need to find out how many words are there in the language in which some pair of letters is the same.

Approach:
To solve this problem, we can use the principle of inclusion-exclusion. First, we can count the total number of words that can be formed using 6 letters without any restriction. Then, we can subtract the number of words in which no letter is repeated. Finally, we can add back the number of words in which two letters are repeated.

Step 1: Counting the total number of words:
There are 6 letters in the language, so there are 6 choices for the first letter, 5 choices for the second letter (as one letter has already been chosen), 4 choices for the third letter, and so on. Therefore, the total number of words that can be formed using 6 letters is:

6 x 5 x 4 x 3 x 2 x 1 = 720

Step 2: Counting the number of words in which no letter is repeated:
To count the number of words in which no letter is repeated, we can use the permutation formula. The number of permutations of n objects taken r at a time is given by:

nPr = n! / (n-r)!

For our problem, n = 6 and r = 6 (as we are choosing all 6 letters). Therefore, the number of words in which no letter is repeated is:

6P6 = 6! / (6-6)! = 720 / 1 = 720

Step 3: Counting the number of words in which two letters are repeated:
To count the number of words in which two letters are repeated, we can choose the two letters that are repeated in 6C2 = 15 ways. Once we have chosen the two letters, we can arrange them in 2! = 2 ways and then arrange the remaining 4 letters in 4! ways. Therefore, the number of words in which two letters are repeated is:

15 x 2 x 4! = 720

Step 4: Apply inclusion-exclusion principle:
We can now apply the inclusion-exclusion principle to count the number of words in which some pair of letters is the same. The number of such words is:

Total number of words - Number of words in which no letter is repeated + Number of words in which two letters are repeated

= 720 - 720 + 720

= 720

Conclusion:
Therefore, there are 720 words in the language in which some pair of letters is the same. This problem can be solved using the principle of inclusion-exclusion, which is a powerful tool for counting problems that involve multiple conditions.

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