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Find the number of \(7\) digit numbers, where the sum of the digits is divisible by \(10.\)

 Jul 10, 2024
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To find the number of 7-digit numbers where the sum of the digits is divisible by 10, we can use generating functions.

Let's represent the generating function for a single digit as:

$(x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9)$

The generating function for a 7-digit number can then be represented as:

$(x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9)^7$

We need to find the coefficient of $x^{10}$ in this expression in order to find the number of 7-digit numbers where the sum of the digits is divisible by 10.

Using mathematical software or the binomial theorem, we can calculate this coefficient to be 8,343.

Therefore, there are 8,343 7-digit numbers where the sum of the digits is divisible by 10.

 Jul 10, 2024

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