Problem 94 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of four digit numbers that can be formed from the digits \(0,1,2,3,4,5\) with at least one digit repeated is (A) 420 (B) 560 (C) 780 (D) None of these

Step-by-Step Solution

Answer

(C) 780

Step 1: Calculate the total number of four-digit numbers

A four-digit number cannot start with 0 as it would then become a three-digit number. Therefore, for the first digit, we have 5 choices: 1, 2, 3, 4, or 5. For the remaining three digits, each can be any of the six digits (0, 1, 2, 3, 4, or 5). Hence, the total number of four-digit numbers that can be formed is calculated as follows: For the first digit, we have 5 choices, and for each of the remaining three digits, we have 6 choices. Therefore, the total number of numbers is:\[5 \times 6 \times 6 \times 6 = 1080\]

Step 2: Calculate the number of four-digit numbers with all different digits

Now, we calculate the number of four-digit numbers where each digit is different. The first digit can be any of the 5 non-zero digits (1, 2, 3, 4, 5). After choosing the first digit, we choose a different digit for the next position, and so on, without repeating any digit out of the 6 available. First digit: 5 choices (as it cannot be 0)Second digit: 5 choices (can be any but the first chosen digit)Third digit: 4 choices (after removing two previously chosen digits)Fourth digit: 3 choices (after removing three previously chosen digits)Therefore, the total number essentially becomes:\[5 \times 5 \times 4 \times 3 = 300\]

Step 3: Calculate numbers with at least one digit repeated

To find the number of four-digit numbers with at least one digit repeated, we subtract the number of all different-digit numbers calculated in Step 2 from the total number of numbers calculated in Step 1.\[1080 - 300 = 780\]

Step 4: Confirm the final answer

The number of four-digit numbers that can be formed using the digits 0, 1, 2, 3, 4, 5 with at least one digit repeated is found to be 780, which matches option (C).

Key Concepts

Permutations

Permutations are a fundamental concept in combinatorics, often used to express the number of different ways to arrange a collection of items. When calculating permutations, it is crucial to consider the order of elements.
Distinct arrangements are counted as separate permutations, which makes the order particularly important.

In the case of selecting numbers, such as in a four-digit sequence, each position in the sequence can be filled by any of the given digits, provided the arrangement rules are followed. For instance, the first digit of a four-digit number cannot be zero, as it would convert the number into a three-digit one. Permutations consider these constraints and help us determine the total possible sequences.

To generate permutations without repetition for our exercise, we must start with the first digit being any of the five non-zero options. The further digits are filled by decrementing choices, ensuring that no digit is repeated. For example, if the first digit is chosen from 5 digits, the second can be any of the remaining 5, the third from 4, and the last from 3, yielding a total of \(5 \times 5 \times 4 \times 3 = 300\) distinct permutations without repetition.

Repetition

Repetition in combinatorics refers to the concept of allowing elements to appear more than once in each arrangement. This is crucial in calculating combinations where repeated elements are permitted.
In the context of the given exercise, repetition allows for larger numbers of arrangements than configurations with unique elements.

To handle repetition, each position in a sequence can be filled by any of the permissible elements, with digit constraints applying as needed. For instance, aside from the first digit constraint explained before, repeated use of any digit for the subsequent positions is allowed.

For our problem, each digit after the first one can include 0, as all six options become available. This leads to multiplying the choices available for each position. Thus, for a four-digit sequence, having 5 initial choices followed by 6 choices for each of the remaining digits results in \(5 \times 6 \times 6 \times 6 = 1080\). The subtraction step helps isolate only those numbers with repetition (at least one digit repeated), where required. This final calculation distinguishes the repeated configurations from all possible ones.

Digit Selection

Digit selection involves choosing specific digits to fit within certain criteria, and it’s a crucial step when constructing number sequences or permutations.
When selecting digits for specific positions, various rules can limit choices, such as restrictions on the starting digit or whether digits can repeat.

In this particular exercise, digit selection begins by choosing any of the non-zero digits for the first position. This restriction helps form a valid four-digit number, as starting with 0 would reduce the number’s length to three digits. Subsequent positions can then fill using any of the given digits, reflecting repeated or distinct digit scenarios as outlined.

Ensuring correct digit selection ensures the valid tally of possible combinations. Each decision leads to a branching of subsequent choices, thus altering how many potential correct combinations we can total. Importantly, the combination setup varies if all distinct digits are needed as opposed to allowing repetitions.