Question

How many four-digit numbers can be formed using the digits \(0,1,2,3,4,5,6,7,8,\) and 9 if the first digit cannot be 0 ? Repeated digits are allowed.

Short answer

9000

Step-by-step solution

Understand the Problem

To determine how many four-digit numbers can be formed using the digits 0 through 9 with the condition that the first digit cannot be 0, and repeated digits are allowed.

Determine the Choice for the First Digit

The first digit of the number can be any digit from 1 to 9 (excluding 0), so there are 9 possible choices for the first digit.

Determine the Choices for the Remaining Digits

Each of the remaining three digits in the four-digit number can be any of the 10 digits (0 through 9). Repeated digits are allowed, so there are 10 possible choices for each of these three digits.

Calculate the Total Number of Combinations

To find the total number of four-digit numbers, multiply the number of choices for each digit position. There are 9 choices for the first digit, and 10 choices for each of the remaining three digits. Thus, the total number of combinations is calculated by: \[ 9 \times 10 \times 10 \times 10 = 9000 \]

Key concepts

Digit Combinations

When dealing with forming numbers, understanding digit combinations is crucial. In our specific problem, we are forming four-digit numbers using digits from 0 to 9. However, since the number is four digits, the first digit cannot be 0. This gives us the range 1 to 9 for the first digit. For the remaining three digits, there are no such restrictions, and any digit from 0 to 9 is permissible. This means we have a broad range of combinations for each place in the number: 9 options for the first digit and 10 for each of the next three. Breaking it down:

• First digit: choices = 9 (1 to 9)
• Second digit: choices = 10 (0 to 9)
• Third digit: choices = 10 (0 to 9)
• Fourth digit: choices = 10 (0 to 9)

By understanding these digit combinations, we can proceed to find the total count of valid four-digit numbers.

Positional Value

Positional value is a fundamental part of understanding numbers. Each digit in a number has a specific place value based on its position. For four-digit numbers, we have places for the thousands, hundreds, tens, and ones digits. For instance, in the number 3527:

• The '3' is in the thousands place, meaning it represents 3000.
• The '5' is in the hundreds place, meaning it represents 500.
• The '2' is in the tens place, meaning it represents 20.
• The '7' is in the ones place, meaning it represents 7.

When forming new numbers, the positional value remains consistent, but what changes are the digits filling these positions. For our problem:

• First position (thousands): must be between 1 and 9.
• Second position (hundreds): can be 0 to 9.
• Third position (tens): can be 0 to 9.
• Fourth position (ones): can be 0 to 9.

Understanding this allows us to logically deduce that each position's options affect the final count.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations. In this problem, we utilize combinatorial principles to find the total number of valid four-digit numbers. We start by identifying the options for each digit as previously discussed. Then, the combinatorial principle tells us to multiply the number of choices for each position. This multiplication rule, fundamental in combinatorics, helps find all possible arrangements without listing each possibility. Here, we have:
\[ 9 \times 10 \times 10 \times 10 = 9000 \]
Therefore, we calculate:

• First digit: 9 choices
• Second digit: 10 choices
• Third digit: 10 choices
• Fourth digit: 10 choices

After multiplying these, we find that there are 9000 unique four-digit numbers satisfying the given conditions. Utilizing combinatorial strategies simplifies complex counting and ensures accuracy in solving such problems.