Investigating numbers with few repeated numbers

how many 18 digit numbers are there such that no digit occurs more than three time in n?

[Stephen23]

@Dnyandeep Sagar: what have you tried so far?

Answer 1

[Amal George M]

The first digit has to be a non-zero digit, and hence it can only be populated 9 ways.

Let a denote the number of times the first digit is repeted in the rest of the 17 possible locations. and () denote the number of times the other digits are repeated in the rest of the 17 possible locations.

Then the number of ways the rest of the locations can be filled is equalvent to the number of solutions to under the constraints and .

This problem can be remodelled as a multiplication problem. Consider the algebraic expression. If we expand this expression, then the coefficent of power of x will be the number of solutions such that sum is n. This can be found by

>> syms x
>> y=((1+x+x^2+x^3)^9*(1+x+x^2));
>> coX=coeffs(y,x);
>> coX(18)
 
        ans =
         
        69834
        

Now, since first place can be filled 9 ways, the final answer willl be 628506.

Correct me if I got this wrong.