How many are these four digit numbers?

There are 165 four-digit numbers with this special property.

Explanation

The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 can be used for the last digit of the number. Zero can be excluded as a sum value because the digit of thousands must not be zero.

For the sum of the value 1, there is only one number that satisfies the condition exactly: 1001.
For the sum of the value 2, there are exactly three numbers that fulfill the condition: 1102; 1012; 2002
For the sum of the value 3, there are exactly six numbers that fulfill the condition: 1113; 1023; 1203; 2103; 2013; 3003
For the sum of the value 4, there are exactly ten numbers that fulfill the condition: 1124; 1214; 1304; 1034; 2114; 2204; 2024; 3104; 3014; 4004

Now numbers with total values ​​​​5, 6, 7, 8 and 9 are missing. Of course, all values ​​\u200b\u200bcan be selected and written individually. The formula for single cumulative values ​​facilitates the solution:

S (x) = 0.5x² + 0.5 times

For example, if you enter the value of the sum of 4 for x, you will get the number of possible four-digit numbers with a value of the sum of 4, ie 10.

The total number of all possible four-digit numbers using this property is obtained by replacing the number 9 in the following formula:

G = (1/6) x³ + (1/2) x² + (1/3) ×