Problem 112 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of integers between 1 and 1000000 that have the sum of the digits 18, is (A) 25927 (B) 25827 (C) 24927 (D) None of these

Step-by-Step Solution

Answer

The answer is (A) 25927.

Step 1: Understand the Problem

We need to determine how many integers between 1 and 1,000,000 have a digit sum of 18. An integer between 1 and 1,000,000 has at most 6 digits.

Step 2: Represent the Integer

Consider a 6-digit number as \( x_1x_2x_3x_4x_5x_6 \), where \( x_1 \) to \( x_6 \) are digits of the number. We have \( x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 18 \) and each \( x_i \) must be a digit (0 to 9).

Step 3: Use Generating Functions

The problem of finding the number of solutions can be translated into a generating function problem. For a digit \( x_i \), the generating function is \( 1 + x + x^2 + \.\.\. + x^9 \), which equals \( \frac{1-x^{10}}{1-x} \).

Step 4: Form the Complete Generating Function

The full generating function for the 6 digits is \( (1 + x + x^2 + ... + x^9)^6 = \left( \frac{1-x^{10}}{1-x} \right)^6 \). We need the coefficient of \( x^{18} \) in this expansion.

Step 5: Simplify and Calculate

Apply the binomial theorem or use computational methods/symmetry to find the polynomial coefficient of \( x^{18} \) after expansion. This is effectively a form of integer partitioning using generating functions.

Step 6: Evaluate and Compare Options

We find through computation that the coefficient of \( x^{18} \) shows 25,927 combinations. Compare this with the options given.

Key Concepts

Generating Functions

Generating functions are a powerful tool in combinatorics for solving problems related to counting. They are particularly useful when you need to find the number of ways to achieve a certain distribution or partition of integers, like summing the digits of a number.

In this problem, each digit of a number can range from 0 to 9. The generating function for a single digit, which represents all possible values it can take, is \( 1 + x + x^2 + \dots + x^9 \), which simplifies to \( \frac{1-x^{10}}{1-x} \).

When you look at multiple digits (up to six, in this case), you take the power of the generating function: \( \left( \frac{1-x^{10}}{1-x} \right)^6 \). The goal is to find the coefficient of \( x^{18} \), representing the sum of 18 across all six digits.

Using generating functions allows you to translate a counting problem into the realm of algebra. It transforms finding solutions into finding coefficients of polynomials, which can be easily handled either analytically or computationally.

Integer Partitioning

Integer partitioning involves finding the number of ways to sum integers in order to reach a specific total. In this exercise, the objective is to partition the number 18 across six non-negative integers, where each integer must be a digit (0 to 9).

To understand why integer partitioning is crucial in this context: consider each digit of a six-digit number that sums to 18. We are effectively trying to distribute 18 units among six containers (digits), but with restrictions as no container can hold more than 9 units.

Using constraints helps in formulating the problem correctly. It's not merely about the partitions of 18, but ensuring that each part is within the limits of digit values, i.e., 0 to 9, following the logic derived with the generating functions. This fusion of partitioning with generates an efficient way to pinpoint possible numbers.

Binomial Theorem

The binomial theorem expands powers of binomials, which makes it an essential tool in finding coefficients of polynomial expressions obtained through generating functions.

In this exercise, the expression \( \left( \frac{1-x^{10}}{1-x} \right)^6 \) contains terms that need expansion to identify the coefficient of \( x^{18} \). The binomial theorem states:\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k \]

Here, it's applied iteratively to expand the generating function for six digits. The complexity comes from calculating such coefficients, which can often be expedited using computational tools. The powerful marriage of generating functions and the binomial theorem lets us compute the exact counts of integer solutions quickly.