Permutations and Combinations: Forming Numbers Greater than 1000

Problem Objective and Constraints

The objective is to determine the total number of distinct integers greater than $1000$ that can be constructed using a specific set of digits. The available digits provided for this task are $\{0, 1, 2, 3\}$. There is a critical constraint placed on the construction of these numbers: the repetition of digits is strictly prohibited. This means that each digit from the set can appear at most once in any single number produced. The goal is to identify which count matches the given options: (a) $24$, (b) $18$, (c) $15$, or (d) $12$.

Identifying the Required Number of Digits

To satisfy the condition of being greater than $1000$, a number must have at least four digits. In this problem, we are restricted to exactly four unique digits: $0$, $1$, $2$, and $3$. Because the repetition of digits is not allowed, the maximum length of any number that can be formed is limited by the size of the set of available digits. Consequently, since we only have four digits and cannot reuse any, the largest possible number we can construct will have exactly four digits. No five-digit or higher numbers can be formed. Therefore, we must calculate the possible permutations of four-digit numbers that can be made using this specific set.

Constraints on the Thousands Place

A four-digit number is comprised of four distinct positional slots: the thousands place, the hundreds place, the tens place, and the units (ones) place. For a sequence of digits to constitute a valid four-digit number, the thousands place (the most significant digit) must not be zero. If the digit $0$ were placed in the thousands position, the number would effectively function as a three-digit number, and any three-digit number formed from the set $\{0, 1, 2, 3\}$ would necessarily be less than the threshold of $1000$.

Given the set of digits $\{0, 1, 2, 3\}$, the possible candidates for the thousands place are limited to:

1. The digit $1$
2. The digit $2$
3. The digit $3$

This results in a total of $3$ possible choices for the first position of the number.

Permutations of the Remaining Positions

After selecting a non-zero digit for the thousands place, there are three remaining positions to fill: the hundreds, tens, and units places. Since the "no repetition" rule is in effect, we have three digits left from the original set of four digits to distribute among these three positions.

The number of ways to arrange these remaining $3$ digits into the $3$ available slots is found by calculating the factorial of the number of items, represented as $n!$. In this specific instance, we calculate the permutations of $3$ items taken $3$ at a time:

$3! = 3 \times 2 \times 1 = 6$

This calculation demonstrates that for every individual choice made for the thousands place, there are $6$ unique ways to complete the remaining digits of the four-digit number.

Calculation of the Total Count

To arrive at the final answer, we apply the fundamental counting principle, multiplying the number of options for the restricted thousands place by the number of permutations available for the subsequent digits.

$\text{Total Numbers} = (\text{Choices for Thousands Place}) \times (\text{Arrangements for Remaining Places})$

Substituting the values derived in the previous steps:

$\text{Total Numbers} = 3 \times 3!$

$\text{Total Numbers} = 3 \times 6 = 18$

Verification and Range Check

The final step is to ensure that all $18$ constructed four-digit numbers are indeed greater than $1000$. The smallest four-digit number that can be formed from the set $\{0, 1, 2, 3\}$ without repetition is created by placing the smallest non-zero digit ($1$) in the thousands place and the remaining digits in ascending order ($0$, $2$, then $3$). This number is $1023$.

Since even the smallest possible non-repeating four-digit number ($1023$) is greater than the target threshold of $1000$, all $18$ permutations are valid. The number $1000$ itself cannot be formed because it would require the repetition of the digit $0$, which is explicitly disallowed. Thus, the total number of integers greater than $1000$ that can be formed using the digits $0$, $1$, $2$, and $3$ without repetition is $18$. This corresponds to option (b).