How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
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Step-by-Step Solution
Step 1: Analyze the given digits and number of places
We are given seven digits: 1, 2, 0, 2, 4, 2, 4. We need to form numbers greater than 1,000,000 using all these digits. A number greater than 1,000,000 must have at least 7 digits. Since we have exactly 7 digits, we must form 7-digit numbers.
Step 2: Calculate total permutations of the digits
First, let's find the total number of distinct 7-digit numbers that can be formed using these digits. We have 7 digits in total, with the digit '2' repeating 3 times and the digit '4' repeating 2 times. The formula for permutations with repetitions is n!/(n1!n2!...), where n is the total number of digits and ni is the frequency of each repeating digit.
Step 3: Calculate permutations where '0' is the first digit
A 7-digit number cannot start with '0'. If '0' is placed in the first position, the number becomes a 6-digit number, which is not greater than 1,000,000. So, we need to subtract these invalid numbers from the total permutations. If '0' is fixed at the first position, we are left with 6 digits: 1, 2, 2, 2, 4, 4. The digit '2' repeats 3 times and '4' repeats 2 times.
Step 4: Subtract invalid permutations from total permutations
To find the number of 7-digit numbers greater than 1,000,000, we subtract the numbers that start with '0' (which are effectively 6-digit numbers) from the total number of permutations of the given digits.