Question

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only is :

A. 77
B. 42
C. 35
D. 82

Answer 1

To find the number of seven-digit integers with a sum of the digits equal to 10, we can use the concept of generating functions. The coefficient of the term x^10 in the polynomial (x + x^2 + x^3)^7 is 35. The correct answer is C. 35.

Step-by-step explanation:

To find the number of seven-digit integers with a sum of the digits equal to 10, we can use the concept of generating functions.

Step 1:

Represent each digit as a term in a polynomial. The polynomial for digits 1, 2, and 3 is (x + x^2 + x^3). This indicates that each digit can appear 0 or more times in the integer.

Step 2:

Raise the polynomial to the power of 7, as we want a seven-digit integer. This is done by multiplying the polynomial by itself 7 times.

Step 3:

Find the coefficient of the term x^10 in the resulting polynomial. This will give us the number of seven-digit integers with a sum of the digits equal to 10.

In this case, the coefficient turns out to be 35, so the answer is C. 35.