Question

How many 3-digit numbers can you form with the numbers 345 if you can use each number just once in each of the 3-digit numbers you create? A. 27 B. 12 C. 9 D. 6 E. 3

Short answer

D. 6

Step-by-step solution

Identify the Available Digits

The digits provided are 3, 4, and 5.

Understand the Restrictions

Each digit can be used only once in each 3-digit number.

Determine the Number of Possible Permutations

Since the digits must be used only once and we are forming a 3-digit number, we need to find the number of permutations of the three digits. The formula for permutations of n distinct objects is given by n!. In this case, n is 3.

Calculate the Permutations

Apply the permutation formula: 3! = 3 × 2 × 1 = 6.

List the Possible 3-Digit Numbers

To double-check, list out all possible permutations: 345, 354, 435, 453, 534, 543.

Choose the Correct Answer

There are 6 possible 3-digit numbers that can be formed. So the correct answer is D. 6.

Key concepts

Permutation Calculation

Permutation calculation is a critical concept in combinatorics.
It allows us to understand how we can arrange a set of items.
When the order matters, we use permutations.
In particular, the formula for permutations of a set of n distinct objects is given by !.This means we multiply all integers from 1 up to n.
In our initial exercise, we had 3 digits: 3, 4, and 5.
Therefore, the number of permutations of these 3 digits is calculated by ! = 3! = 3 × 2 × 1 = 6.
This result tells us there are 6 different ways to arrange these 3 numbers, confirming our answer.

Three-Digit Numbers

To form a three-digit number, each digit must be distinct and used only once.
This ensures that the numbers are true permutations.
In the example problem, we formed three-digit numbers using the digits 3, 4, and 5.
The possible arrangements are: 345, 354, 435, 453, 534, and 543.
Notice each number uses each of the three digits precisely once, maintaining our restriction.
Comprehending this concept of unique usage is important for realizing how the permutations play out in this scenario.

Mathematical Problem-Solving

Approaching mathematical problems systematically is the key to finding accurate solutions.
First, identify all given elements and imposed restrictions, just like we identified the digits 3, 4, and 5.
Next, understand what the problem is asking.
In this case, it was asking to find the number of unique three-digit numbers.
Then, apply the appropriate mathematical formula or process.
Here, we used the permutation formula ! to find our answer.
Finally, verify your findings by listing possible outcomes.
Listing the permutations of 345, 354, 435, 453, 534, and 543 confirmed that exactly 6 unique three-digit numbers could be formed.