Question

The number of seven-digit positive integers formed using the digits 1, 2, 3, and 4 only, and whose sum of the digits is 12, is         

• A. 413

Answer 1

The Correct Option is A

The problem requires us to find the number of seven-digit positive integers formed using the digits 1, 2, 3, and 4 only, such that the sum of the digits is 12.

To approach this problem, we use a combinatorial method known as the "stars and bars" method, along with generating functions.

Each possible digit position in the seven-digit number can be occupied by one of the numbers 1, 2, 3, or 4. Let us denote the number of times each digit appears in the number by \(x_1, x_2, x_3, x_4\) for the digits 1, 2, 3, and 4 respectively. Therefore, we have the equation:

\(x_1 + x_2 + x_3 + x_4 = 7\) (since the total number of digits is 7)

We also have the condition for the sum of the digits:

\(x_1 \times 1 + x_2 \times 2 + x_3 \times 3 + x_4 \times 4 = 12\)

Now, we need to solve these under the constraint that each \(x_i \geq 0\).

We employ generating functions. The generating function for each digit is:

\(g(x) = x^1 + x^2 + x^3 + x^4\)

This is simplified to:

\(g(x) = x(1 + x + x^2 + x^3)\)

This represents the choices for each digit. Thus, for a seven-digit number, the generating function for the entire expression is:

\((x^1 + x^2 + x^3 + x^4)^7\)

We are interested in the coefficient of \(x^{12}\) in the expansion of:

\((x^1 + x^2 + x^3 + x^4)^7\)

Using the multinomial theorem, you expand this and find it corresponds to 413 such numbers, which precisely matches one of the given options.

Therefore, the correct answer is 413.