Problem 117 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of numbers greater than \(10^{6}\) that can be formed using the digits of the number 2334203 , if all the digits of the given number must be used, is (A) 360 (B) 420 (C) 260 (D) None of these

Step-by-Step Solution

Answer

The answer is 360.

Step 1: Identify the Digits

The number given is 2334203, which contains the digits 2, 3, 3, 4, 2, 0, and 3. Count how many times each digit occurs. We have: - Two 2s - Three 3s - One 4 - One 0

Step 2: Calculate Total Permutations

To find the total number of permutations of these digits, divide the factorial of the total number of digits by the factorial of the number of repetitions of each digit. - Total digits: 7 - Repeating digits: Two 2s and three 3s The formula is: \[ \frac{7!}{2! \cdot 3!} \] Calculating this gives: \[ \frac{5040}{2 \cdot 6} = \frac{5040}{12} = 420 \] total permutations of the digits.

Step 3: Exclude Permutations Starting with '0'

Any number greater than \(10^6\) cannot start with digit '0'. First, calculate permutations that start with '0' and then subtract from total permutations. Permutations starting with '0': - Remaining digits: 2, 3, 3, 4, 2, 3 - Total permutations without leading '0' (6 digits): \[ \frac{6!}{2! \cdot 3!} = \frac{720}{12} = 60 \] So, there are 60 numbers that start with '0'.

Step 4: Calculate Valid Permutations

Subtract the permutations starting with '0' from the total permutations to find valid numbers: \[ 420 - 60 = 360 \]

Key Concepts

Factorial

In mathematics, the concept of a factorial is fundamental when dealing with permutations and combinations. A factorial, represented by the symbol "!", is the product of all positive integers up to a given number. For example, 7! (read as "seven factorial") is calculated as follows:\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]Factorials are crucial in determining how many different ways elements can be arranged. This is because the number of possible permutations of a set of items is the factorial of the number of items, assuming no repetition of elements. For the problem here, we calculate the factorial of the total number of digits in the sequence, which is 7!.The use of factorials extends beyond basic sequence arrangement. They are used in statistical formulas, probability calculations, and even in solving algebraic equations involving permutations and combinations.

Digits

Understanding and identifying the digits involved in any permutation problem is a key step. In our example, the given number is 2334203. It comprises the digits 2, 3, 3, 4, 2, 0, and 3, each contributing to the formation of new permutations. When we analyze these digits, it's important to note how often each digit appears:

• The number 2 appears twice.
• The number 3 appears three times.
• The number 4 appears once.
• The number 0 appears once.

By recognizing the frequency of each digit, we enable accurate calculation of permutations, especially when considering repetitions.Moreover, in problems where forming numbers is key, the value of digits and their arrangement can drastically change the properties and the possible range of numbers generated. In our scenario, ensuring that the resulting number exceeds \(10^6\) is especially important, which restricts starting the sequence with the digit '0'. This step is critical as leading zeros can reduce the numerical value significantly.

Repetition in permutations

Repetition in permutations is an integral concept when identifying the number of valid permutations involving non-unique elements. When digits repeat, the total number of permutations reduces.For instance, if all the elements were unique, the permutation count would strictly be \( n! \), where \( n \) is the number of digits. However, with repeating digits, such as in the number 2334203, adjustments are necessary. This is done by dividing the total factorial by the factorial of each set of identical digits:\[ \frac{7!}{2! \cdot 3!} \]Here, 2! accounts for the two 2's, and 3! for the three 3's. This division corrects over-counting where identical digits are mistakenly treated as unique during permutation calculations.The rationale behind dividing by factorials of identical items is due to each identical item being indistinguishable among themselves – rearranging them does not create a new permutation. Thus, understanding repetition allows us to generate accurate counts of distinct number arrangements from a set of digits.