Problem 95 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of odd numbers lying between 40000 and 70000 that can be made from the digits \(0,1,2,4,5,7\) if digits can be repeated in the same number is (A) 864 (B) 932 (C) 766 (D) None of these

Step-by-Step Solution

Answer

The number of odd numbers is none of the options; it should be 1296.

Step 1: Understanding the Problem

We need to determine how many odd numbers can be formed using the digits 0, 1, 2, 4, 5, and 7 while keeping them between 40000 and 70000. Also, these numbers should be odd, which implies that their units digit must be an odd digit.

Step 2: Identifying the Units Digit

Since the number must be odd, the possible choices for the units digit are odd digits from the set {0, 1, 2, 4, 5, 7}, i.e., 1, 5, and 7.

Step 3: Formulating the Constraints

The number should be between 40000 and 70000, which means the ten-thousands digit must be either 4, 5, or 6. However, since 6 is not part of our allowed digits, the ten-thousands digit can only be 4 or 5.

Step 4: Calculating Combinations

For five-digit numbers:1. Units digit: 3 choices (1, 5, 7).2. Ten-thousands digit: 2 choices (4, 5).3. Thousands, hundreds, and tens digits: Each of these can be any of the six digits (0, 1, 2, 4, 5, 7). Each place has 6 choices.Thus, the total number of combinations is given by the expression: \[ 2 \text{ (Ten-thousands digits)} \times 6^3 \text{ (Thousands, Hundreds, Tens choices)} \times 3 \text{ (Units digits)} \]

Step 5: Computing the Total Number of Combinations

Now, we compute the total:\[ 2 \times 6^3 \times 3 = 2 \times 216 \times 3 = 2 \times 648 = 1296 \]

Key Concepts

Odd numbers

When dealing with numbers, an odd number is any integer that cannot be evenly divided by 2. When creating odd numbers from a set of digits, it's crucial to focus on the unit's digit.
This is because only when the unit's digit is odd that the whole number becomes odd.

• The odd digits from the set \( \{0, 1, 2, 4, 5, 7\} \) used in this exercise are 1, 5, and 7.

Therefore, when forming numbers, ensuring the unit's digit falls within these choices guarantees that the formed number is odd. This is a fundamental step in combinatorial problems involving the formation of odd numbers.

Digit repetition

Digit repetition allows using the same digit more than once in a number formation.
In the exercise, each digits from the set \( \{0, 1, 2, 4, 5, 7\} \) can be used multiple times. This increases the number of combinations significantly.

• For instance, even if some digits are already used as part of a number, they can be reused in different positions.
• This characteristic is crucial for combinatorial calculations as it keeps the potential set of choices for each digit in the formation unchanged.

This concept is particularly useful when constraints are placed on numeric combinations, such as ensuring the number remains odd while remaining within a specified range.

Number range constraints

Number range constraints restrict the possible numbers within a certain boundary. In the given exercise, the numbers must lie between 40000 and 70000.
This restriction impacts the possible choices for each digit, especially for the ten-thousands digit.

• The ten-thousands digit must either be 4 or 5 because 6 is not among the available digits.
• This constraint guides the calculation and ensures that the resulting numbers fit within the specified range.

By understanding these constraints, one can strategically limit and guide the choice of combinations during number formation, making sure all created numbers adhere to all specified rules, ensuring they are both valid and countable.