We are given a set of digits: $3, 4, 5, 6, 6, 7, 7, 8$ and we are tasked with finding how many different integers $n$ can be formed such that $n > 60,00,000$ (6 million).

Step 1: Number of digits

Since the number $n$ must exceed 6 million, $n$ must be at least 7 digits long. Therefore, $n$ will be a 7-digit number or an 8-digit number.

Step 2: Counting the 7-digit numbers

To form a 7-digit number, we must select and arrange 7 digits from the set of available digits.

Case 1: First digit is 6, 7, or 8

For $n > 6,000,000$, the first digit must be either 6, 7, or 8. Let's consider each case:

• First digit is 6: We are left with the digits $3, 4, 5, 6, 7, 7, 8$. These 7 digits include a repetition of the digit 7, so the number of distinct arrangements of these digits is: $\frac{7!}{2!} = \frac{5040}{2} = 2520$

• First digit is 7: We are left with the digits $3, 4, 5, 6, 6, 7, 8$. These 7 digits include a repetition of the digit 6, so the number of distinct arrangements is: $\frac{7!}{2!} = \frac{5040}{2} = 2520$

• First digit is 8: We are left with the digits $3, 4, 5, 6, 6, 7, 7$. These 7 digits include repetitions of both 6 and 7, so the number of distinct arrangements is: $\frac{7!}{2!2!} = \frac{5040}{4} = 1260$

The total number of 7-digit numbers is: $2520 + 2520 + 1260 = 6300$

Step 3: Counting the 8-digit numbers

All 8 digits can be used to form an 8-digit number. The number of distinct arrangements, considering the two 6's and the two 7's, is: $\frac{8!}{2!2!} = \frac{40320}{4} = 10080$

Step 4: Total number of valid numbers

The total number of integers $n$ that exceed 6 million is the sum of the 7-digit and 8-digit numbers: $6300 + 10080 = 16380$

Thus, the total number of positive integers $n$ that can be formed using the digits $3, 4, 5, 6, 6, 7, 7, 8$ and exceed 60,00,000 is 16,380.

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Do you want details on any step or have any questions?

Here are 5 related questions:

1. How would the count change if repetitions of 6 and 7 were not allowed?
2. What if $n$ had to be less than 60,00,000 instead of exceeding it?
3. How many 7-digit numbers can be formed if only even digits are allowed in the first place?
4. How does the formula change if there were 3 repeated digits instead of 2?
5. What if the first digit could not be 6, 7, or 8?

Tip: When arranging objects with repeated elements, always account for these repetitions by dividing by the factorial of the number of repeated objects.