How many 7 digit positive integers are there such that the product of the individual digits of each number is equal to 10000?
To find the number of 7-digit positive integers whose digits multiply to 10000, we need to break down the problem into smaller steps.
Step 1: Find the prime factorization of 10000.
10000 = 2^4 * 5^4
Step 2: Determine the number of ways to distribute the powers of 2 and 5 among the seven digits.
2^4 can be distributed among the seven digits in (7+4-1) choose (4) ways, by considering stars and bars.
Similarly, 5^4 can be distributed among the seven digits in (7+4-1) choose (4) ways.
Therefore, the total number of ways to distribute the powers of 2 and 5 among the seven digits is [(7+4-1) choose (4)] * [(7+4-1) choose (4)].
Step 3: Determine the number of ways to distribute the remaining two digits.
Since the remaining two digits can be any non-zero digit, there are 9 choices for each digit.
Therefore, the total number of ways to distribute the remaining two digits is 9^2.
Step 4: Multiply the results from steps 2 and 3 to get the final answer.
The total number of 7-digit positive integers whose digits multiply to 10000 is [(7+4-1) choose (4)] * [(7+4-1) choose (4)] * 9^2.
Calculating the expression [(7+4-1) choose (4)] * [(7+4-1) choose (4)] * 9^2 gives us the answer.