Problem 9
Question
The number of five-digit numbers that contain 7 exactly once is a. \((41)\left(9^{3}\right)\) b. \((37)\left(9^{3}\right)\) c. (7) \(\left(9^{4}\right)\) d. \((41)\left(9^{4}\right)\)
Step-by-Step Solution
Verified Answer
The answer is (7) \(\left(9^4\right)\).
1Step 1: Understand the Problem
We need to find the total number of five-digit numbers where the digit 7 appears exactly once.
2Step 2: Consider the Positions for Digit 7
A five-digit number has positions: thousands, ten-thousands, hundreds, tens, and units. Choose one position for the digit 7. There are 5 such positions.
3Step 3: Determine Digits for Other Positions
For the remaining 4 positions, choose from any digit between 0 and 9 except 7. For each of these positions, there are 9 possible choices.
4Step 4: Calculate the Total Number of Combinations
Multiply the number of positions where 7 can be placed by the number of choices for each of the other 4 digits. This yields: \(5 \times 9^4\).
5Step 5: Check for Matching Options
Examine the options given and find the matching expression for the calculated one.
Key Concepts
Five-Digit NumbersDigit PositioningPermutationsExact Occurrence
Five-Digit Numbers
When we think about creating a five-digit number, it's essential to understand what this means in context. A five-digit number is composed of five consecutive digits, with the first digit being non-zero so that the number is valid and significant.
Five-digit numbers start from 10,000 and go up to 99,999. These numbers can range across different positional values, which are thousands, ten-thousands, hundreds, tens, and units. These five positions are where the specific digits are assigned in the number.
In this problem, since we are focusing on numbers where '7' appears exactly once, it becomes essential to plan the numbering in a way that it adheres to the constraints of being a five-digit formation.
Five-digit numbers start from 10,000 and go up to 99,999. These numbers can range across different positional values, which are thousands, ten-thousands, hundreds, tens, and units. These five positions are where the specific digits are assigned in the number.
In this problem, since we are focusing on numbers where '7' appears exactly once, it becomes essential to plan the numbering in a way that it adheres to the constraints of being a five-digit formation.
Digit Positioning
In a five-digit number, each digit has a specific position that affects the number's value. These positions are ordered as follows:
For our problem, we want '7' to appear exactly once. Thus, we have to select one of these five available positions to place the digit '7'. Therefore, there are five potential ways to choose the position for the digit '7'. This restricted placement forms a foundation for further permutations of the remaining digits.
- Ten-thousands place
- Thousands place
- Hundreds place
- Tens place
- Units place
For our problem, we want '7' to appear exactly once. Thus, we have to select one of these five available positions to place the digit '7'. Therefore, there are five potential ways to choose the position for the digit '7'. This restricted placement forms a foundation for further permutations of the remaining digits.
Permutations
Permutations are arrangements or sequences of objects in a specific order. In our exercise, we are determining how the available digits can fill the remaining positions once the digit '7' is set.
With '7' occupying one position, the other four positions need to be filled with digits from 0-9, excluding '7'. This gives us 9 options (all digits except for '7') for each of the remaining 4 positions, leading to a permutation calculation.
With '7' occupying one position, the other four positions need to be filled with digits from 0-9, excluding '7'. This gives us 9 options (all digits except for '7') for each of the remaining 4 positions, leading to a permutation calculation.
- For each of the 4 positions: 9 choices of digits.
Exact Occurrence
The concept of exact occurrence emphasizes that a specific event (or digit) appears precisely a certain number of times in a given context. Here, our goal is to ensure that '7' appears exactly once in the five-digit number.
Achieving exact occurrence involves strict counts and placements. By having '7' once, we essentially fix it in a single position and ensure no other occurrences in the number. This requirement makes the initial choice of position critical.
Once the position is chosen, the remaining positions are adjusted accordingly to satisfy this condition. The methodology to account for exact occurrence often involves systematically placing the specific digit in different positions and calculating permutations for the balance.
Achieving exact occurrence involves strict counts and placements. By having '7' once, we essentially fix it in a single position and ensure no other occurrences in the number. This requirement makes the initial choice of position critical.
Once the position is chosen, the remaining positions are adjusted accordingly to satisfy this condition. The methodology to account for exact occurrence often involves systematically placing the specific digit in different positions and calculating permutations for the balance.
Other exercises in this chapter
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