Problem 24 - Precalculus with Limits

Question

How many four-digit numbers can be formed under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be less than \( 5000 \). (d) The leading digit cannot be zero and the number must be even.

Step-by-Step Solution

Answer

(a) 9000 (b) 4536 (c) 4000 (d) 4500

Step 1: Solve part (a)

Since the first digit cannot be 0, there would be 9 possible numbers for the first digit (from 1 to 9). However, for the remaining three places, each can have any of the 10 digits from 0 through 9. Therefore, the number of 4-digit numbers that can be formed in this case by the counting principle is \( 9 * 10 * 10 * 10 = 9000 \)

Step 2: Solve part (b)

In this case, not only can the first digit not be 0, there should also be no repetition of digits. For the first digit, it could be any from 1-9 (which gives 9), for the second it can be any except the first (which gives 9) and for the third and fourth digits, they would be any number other than the previous ones (which therefore gives 8 and 7 respectively). Therefore, the number of 4-digit numbers that can be formed in this case is \( 9 * 9 * 8 * 7 = 4536 \)

Step 3: Solve part (c)

For the number to be less than 5000, the first digit could only be from one to four (4 possibilities). However, the other three places can have any of the 10 digits from 0 through 9. Therefore, the number of 4-digit numbers that can be formed in this case is \( 4 * 10 * 10 * 10 = 4000 \)

Step 4: Solve part (d)

Here, the condition is that the number must be even. For a number to be even, the last digit can be 0, 2, 4, 6, 8 (5 possibilities). As usual, the first digit could be any from 1-9 and remaining one place can have any of the 10 digits from 0 through 9. Therefore, the number of 4-digit numbers that can be formed in this case is \( 9 * 10 * 10 * 5 = 4500 \)

Key Concepts

Four-Digit Numbers

Four-digit numbers are numerical values composed of four digits, and they range from 1000 to 9999 inclusively. To form a four-digit number, the first digit cannot be zero, since that would result in a three-digit number. This restriction applies regardless of any additional constraints given by an exercise or problem. Here are some quick facts about four-digit numbers:

• The smallest four-digit number is 1000.
• The largest four-digit number is 9999.
• There are 9,000 possible combinations if we consider all possible numbers from 1000 to 9999.

Understanding how these numbers are structured is crucial when we apply different constraints, such as in problems involving restrictions and the counting principle.

Digit Restriction

Digit restriction is a vital aspect of combinatorial problems where certain criteria limit the choice of digits. In exercises like these, we often see constraints like the leading digit including important rules or conditions. These constraints are put in place to meet specific requirements or limitations. Let's go over some common types:

• Leading digit restrictions like ensuring it cannot be zero, to maintain a four-digit format.
• Constraints that prevent repetitions of digits, which limits the option for each subsequent digit.
• Additional conditions like the requirement for the entire number to be even or within a certain numerical range.

Recognizing these restrictions is key to accurately calculating the total number of eligible digit combinations. It narrows down possibilities to suit the given conditions, thereby providing solutions that meet the specified constraints.

Counting Principle

The counting principle is a fundamental rule in combinatorics employed to find the number of possible outcomes by multiplying the number of choices for each event. It provides a straightforward way to compute the number of ways events can occur, especially useful in forming numbers with set conditions. In easier terms, it's about 'how many options' you have at each step or place in the number. For example:

• If selecting a non-zero leading digit (1-9) implies 9 options.
• Subsequent digits, with no restrictions (0-9), offer 10 options each.
• The product of these available choices gives the total number of possibilities.

When applying the counting principle, outline each place in the number, consider any limitations, and multiply the possible selections for each to arrive at the full count. The principle makes large counts manageable and ensures all conditions are consistently factored in.

Number Properties

Number properties refer to the characteristics and features of numbers that influence how they can be arranged or manipulated under certain conditions. When tackling problems involving forming four-digit numbers, understanding relevant number properties helps in applying the correct rules or conditions. Let's explore some key properties:

• Even Numbers: Numbers whose last digit is 0, 2, 4, 6, or 8 are considered even. This property is crucial when the problem demands an even number.
• Number Range: Defining a number as less than a specific limit (e.g., 5000) requires that we appropriately narrow the first digit's options in our combinations.
• Digit Uniqueness: In scenarios where repetition isn't allowed, each digit must remain distinct, necessitating careful calculation of possible choices at each digit's placement.

Using these properties ensures that designed numbers conform to the problem's requirements, contributing to methodical and accurate solution strategies.