Two 4-digit numbers are to be formed such that the sum of the number is also a 4-digit number and in no place the addition is with carrying the number of ways of forming the number under above conditions is
I am taking it that you mean the sum of the two 4 digit numbers must be a 4 digit number and in no column can there be a carry.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.