Calculating 5-Digit Numbers with Distinct Digits

The problem asks us to find the total number of 5-digit numbers that can be formed using distinct digits selected from the set of digits $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$.

A 5-digit number has five places: ten thousands, thousands, hundreds, tens, and units. Since the digits must be distinct, each digit used must be different from the others.

We have 10 distinct digits available: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

A key constraint for a number to be a 5-digit number is that the digit in the ten thousands place (the first digit) cannot be 0.

Method 1: Filling Positions Step-by-Step

Let's consider the number of choices for each position:

• Ten Thousands Place (First Digit): This digit cannot be 0. Therefore, there are 9 possible choices for this position (digits 1 through 9).
• Thousands Place (Second Digit): We have already used one digit for the first position. We have 9 remaining digits (the 0 is now available, plus the remaining 8 non-zero digits). So, there are 9 possible choices for this position.
• Hundreds Place (Third Digit): We have used two distinct digits for the first two positions. From the original 10 digits, 8 digits remain. So, there are 8 possible choices for this position.
• Tens Place (Fourth Digit): We have used three distinct digits. 7 digits remain. So, there are 7 possible choices for this position.
• Units Place (Fifth Digit): We have used four distinct digits. 6 digits remain. So, there are 6 possible choices for this position.

Using the multiplication principle, the total number of distinct 5-digit numbers is the product of the number of choices for each position:

Number of 5-digit numbers = (Choices for 1st digit) × (Choices for 2nd digit) × (Choices for 3rd digit) × (Choices for 4th digit) × (Choices for 5th digit)

Number of 5-digit numbers = $9 \times 9 \times 8 \times 7 \times 6$

Let's calculate this value:

So, the total number of 5-digit numbers with distinct digits is 27,216.

Method 2: Using Permutations with Restrictions

Another way to approach this problem is to first calculate the total number of permutations of 5 distinct digits chosen from 10 digits and then subtract the number of cases that are not valid 5-digit numbers (i.e., those starting with 0).

Step 1: Total Permutations of 5 Distinct Digits from 10

The number of ways to arrange 5 distinct digits chosen from 10 is given by the permutation formula $\text{P}(n, k) = \frac{n!}{(n-k)!}$, where $n$ is the total number of items (10 digits) and $k$ is the number of items to choose and arrange (5 digits).

Total permutations of 5 distinct digits from 10 = $\text{P}(10, 5) = \frac{10!}{(10-5)!} = \frac{10!}{5!}$

$\text{P}(10, 5) = 10 \times 9 \times 8 \times 7 \times 6 = 30240$

This counts all sequences of 5 distinct digits, including those that start with 0.

Step 2: Permutations that Start with 0 (Invalid 5-Digit Numbers)

If a 5-digit number starts with 0, the first digit is fixed as 0. We then need to choose and arrange the remaining 4 distinct digits from the remaining 9 digits (1 through 9).

The number of ways to choose and arrange 4 distinct digits from the remaining 9 is given by $\text{P}(9, 4)$.

Permutations starting with 0 = $\text{P}(9, 4) = \frac{9!}{(9-4)!} = \frac{9!}{5!}$

$\text{P}(9, 4) = 9 \times 8 \times 7 \times 6 = 3024$

These are the numbers that look like 0xxxx, which are not 5-digit numbers.

Step 3: Subtract Invalid Permutations from Total Permutations

The number of valid 5-digit numbers with distinct digits is the total number of permutations of 5 distinct digits from 10 minus the number of permutations that start with 0.

Number of valid 5-digit numbers = Total Permutations - Permutations starting with 0

Number of valid 5-digit numbers = $\text{P}(10, 5) - \text{P}(9, 4) = 30240 - 3024$

$30240 - 3024 = 27216$

Conclusion on Distinct 5-Digit Numbers

Both methods yield the same result. The total number of 5-digit numbers that can be composed of distinct digits from 0 to 9 is 27,216.

Revision Table: Key Concepts in Counting Distinct Numbers

Additional Information: Counting Problems and Constraints

Counting problems, especially those involving permutations and combinations, often involve constraints. It is crucial to identify these constraints before applying formulas.

• Constraints on Position: Like in this problem, where the first digit cannot be 0. Address the constrained positions first.
• Constraints on Digits/Items: Are digits distinct? Are specific digits required or excluded? Are there limits on the value of the number (e.g., even, odd, greater than X)?
• Understanding the Difference: Know when to use permutations (order matters) versus combinations (order does not matter). In number formation, the order of digits matters, so permutations are relevant.
• Break Down the Problem: Complex counting problems can often be solved by breaking them into smaller, manageable steps, as shown in Method 1.

Practicing various types of counting problems with different constraints helps build proficiency in combinatorics.