Problem 106 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of 7 digit numbers the sum of whose digits is even, is (A) \(35 \times 10^{5}\) (B) \(45 \times 10^{5}\) (C) \(50 \times 10^{5}\) (D) None of these

Step-by-Step Solution

Answer

The answer is (B) \(45 \times 10^5\).

Step 1: Understanding the Exercise

We need to find the number of 7-digit numbers such that the sum of their digits is even. A 7-digit number ranges from 1,000,000 to 9,999,999.

Step 2: Calculating Total 7-digit Numbers

A 7-digit number can have digits from 1 to 9 in the first position, and 0 to 9 in the other six positions. Thus, the total number of 7-digit numbers is given by: \[ 9 \times 10^6 \] since there are 9 choices for the first digit and 10 choices for each of the remaining six digits.

Step 3: Understanding Even and Odd Sums

Consider that each digit from 0 to 9 contributes to the sum being either even or odd. For a number to have an even sum of digits, it is as likely as having an odd sum of digits since the digits range symmetrically around odd and even numbers.

Step 4: Finding Numbers with Even Digit Sum

Since sums being even or odd are equally likely, half of all 7-digit numbers will have an even sum of digits. Therefore, the number of such numbers is half of the total number of 7-digit numbers: \[ \frac{9 \times 10^6}{2} = 4.5 \times 10^6 \]

Step 5: Converting to the Given Choices

The result \(4.5 \times 10^6\) is equivalent to \(45 \times 10^5\), which matches: (B) \(45 \times 10^5\).

Key Concepts

Understanding 7-digit numbers

A 7-digit number is any integer number that has exactly seven digits, ranging from 1,000,000 to 9,999,999. To construct such numbers, certain rules about digits must be followed:

• The first digit cannot be zero, as that would result in a number with fewer than seven digits. Therefore, the first digit ranges from 1 to 9, providing 9 possible choices.
• Each subsequent digit can be any number from 0 to 9, offering 10 possible choices per position.

Consequently, to calculate the total number of 7-digit numbers, you consider the choices for each digit:

• First digit: 9 choices (1 to 9)
• Each of the next six digits: 10 choices (0 to 9)

Using the principle of permutations, the total number of 7-digit numbers is given by multiplying these possibilities together: \[ 9 \times 10^6 \] This calculates to 9 million possible 7-digit numbers, providing a vast range from which to explore other properties.

Calculating Even Sum of Digits

The sum of the digits in a number can be even or odd. By definition:

• An even number is divisible by 2.
• An odd number is not divisible by 2.

To find how many of the 7-digit numbers have a sum of digits that is even, consider the distribution of even and odd sums. Since each digit from 0 to 9 contributes equally to forming sums, whether the total sum is even or odd has a 50-50 probability. Therefore, half of the total 7-digit numbers will have an even sum of digits. Given there are 9 million 7-digit numbers, the count of numbers with an even sum of digits is:\[ \frac{9 \times 10^6}{2} = 4.5 \times 10^6 \] This simple probability approach allows us to ascertain that precisely half, or 4.5 million of these numbers, will have digit sums that are even.

Permutations and Combinations in Combinatorics

Combinatorics is a branch of mathematics dealing with permutations and combinations, among other things. It revolves around counting, arranging, and selecting objects according to certain constraints, which is fundamental to various problems including this one.

**Permutations** refer to arrangements where order matters. Here, when forming 7-digit numbers, each digit's position significantly affects the number generated.

**Combinations**, on the other hand, involve selection where order doesn't matter. While combinations aren’t directly used in this exercise, understanding this allows for grasping deeper probabilistic and counting methods.

In our scenario, permutations of digits give rise to different number configurations leading to a thorough understanding of possible outcomes. By dividing the total permutations, as seen in our calculation of even and odd sums, we apply the principles of equal probability under permutations to simplify the complex counting involved in combinatorics.