To find the number of seven-digit integers with a sum of digits equal to 10 and formed by using the digits 1, 2, and 3 only, we can use the concept of generating functions.

Generating Functions:
A generating function is a formal power series that encodes information about a sequence of numbers. In this case, we can represent the possible digits (1, 2, and 3) as terms in a generating function and use it to find the coefficient of the desired term.

Step 1: Represent the possible digits as terms in the generating function.
Let's represent the digits 1, 2, and 3 as x, x^2, and x^3 respectively.

Step 2: Define the generating function.
The generating function for the sum of digits can be represented as:
G(x) = (x + x^2 + x^3)^7

Step 3: Find the coefficient of the term with x^10.
To find the coefficient of the term with x^10, we need to expand the generating function and look for the term with the desired exponent.

Step 4: Expand the generating function.
Expanding G(x) = (x + x^2 + x^3)^7 using the binomial theorem, we get:
G(x) = (x^7)(1 + x + x^2)^7

Step 5: Find the coefficient of the term with x^10.
To find the coefficient of the term with x^10, we need to find all the ways to select the powers of x^1, x^2, and x^3 that add up to 10. This can be done by finding the coefficient of x^10 in the expansion of (1 + x + x^2)^7.

Step 6: Expand (1 + x + x^2)^7.
Expanding (1 + x + x^2)^7 using the binomial theorem, we get:
(1 + x + x^2)^7 = C(7, 0) + C(7, 1)x + C(7, 2)x^2 + C(7, 3)x^3 + C(7, 4)x^4 + C(7, 5)x^5 + C(7, 6)x^6 + C(7, 7)x^7

Step 7: Find the coefficient of x^10.
The coefficient of x^10 is the coefficient of the term C(7, 3)x^3 in the expansion of (1 + x + x^2)^7. C(7, 3) represents the number of ways to choose 3 elements from a set of 7, which is equal to 35.

Therefore, the number of seven-digit integers with a sum of digits equal to 10 and formed by using the digits 1, 2, and 3 only is 35.