To find the quantity of four-digit numbers that can be created by utilizing the digits from 1 to 9 when repetition of digits is not allowed, we need to understand the concept of permutations. A permutation is an arrangement of items where the order matters.

Understanding Four-Digit Numbers Formation

We are tasked with forming a four-digit number using a set of 9 distinct digits (1, 2, 3, 4, 5, 6, 7, 8, 9). Since repetition of digits is not allowed, each digit used for a specific place value (thousands, hundreds, tens, units) must be unique.

• Thousands Place: For the first digit of the four-digit number (the thousands place), we have 9 available digits (from 1 to 9). So, there are 9 choices.
• Hundreds Place: After selecting a digit for the thousands place, we cannot repeat it. This means we have 8 digits remaining for the hundreds place. So, there are 8 choices.
• Tens Place: Similarly, after selecting digits for the thousands and hundreds places, we are left with 7 distinct digits for the tens place. So, there are 7 choices.
• Units Place: Finally, for the units place, we have 6 remaining distinct digits. So, there are 6 choices.

Calculating the Quantity of Numbers

The total number of four-digit numbers that can be formed is found by multiplying the number of choices for each position. This is a direct application of the permutation formula $P(n, r)$, where $n$ is the total number of items to choose from, and $r$ is the number of items to choose and arrange.

In this problem:

• Total number of digits available ($n$) = 9
• Number of digits to be chosen for the four-digit number ($r$) = 4

The formula for permutations is given by:

$\text{P}(n, r) = \frac{n!}{(n-r)!}$

Substituting the values of $n=9$ and $r=4$ into the permutation formula:

$\text{P}(9, 4) = \frac{9!}{(9-4)!} = \frac{9!}{5!}$

Expanding the factorials, we get:

$\text{P}(9, 4) = 9 \times 8 \times 7 \times 6 \times \frac{5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1}$

$\text{P}(9, 4) = 9 \times 8 \times 7 \times 6$

Step-by-step Multiplication for the Quantity:

• First, multiply the choices for the first two places: $9 \times 8 = 72$
• Next, multiply this result by the choice for the third place: $72 \times 7 = 504$
• Finally, multiply this result by the choice for the fourth place: $504 \times 6 = 3024$

Therefore, the quantity of four-digit numbers that can be created by utilizing the digits from 1 to 9, without repetition, is 3024.

Summary of Results for Four-Digit Numbers