How many different numbers of two digits can be formed with the digits 1, 2, 3, 4, 5, 6; no digit being repeated?

Question: How many different numbers of two digits can be formed with the digits 1, 2, 3, 4, 5, 6; no digit being repeated?

Choices:

• 30

• 35

• 40

• 45

Correct Answer: 30

<h2>Reasoning:</h2>

To determine how many different two-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 without repeating any digit, we can approach the problem step-by-step:

<h3>Step 1: Choosing the Tens Place</h3>

There are 6 choices for the tens place: 1, 2, 3, 4, 5, and 6. Since the digit chosen for the tens place must be unique and used only once, each choice reduces the number of available digits for the units place by 1.

<h3>Step 2: Choosing the Units Place</h3>

After a digit has been chosen for the tens place, we have 5 remaining choices for the units place.

<h3>Step 3: Calculating the Total Combinations</h3>

Since the choice for each position is independent of the other, we use multiplication to find the total number of possible combinations:

$$ 6 \text{ choices for tens place} \times 5 \text{ choices for units place} = 30 \text{ different two-digit numbers} $$

<h3>Conclusion:</h3>

Thus, the correct answer is 30. This can be verified by listing out all possible combinations, but the formula used confirms the correct count effectively and efficiently.

<h2>Additional Notes:</h2>

It's always useful to understand the underlying principles of combination and permutation problems:

• The absence of repetition constraints simplifies the calculation.

• The order of selection (tens and units place) matters in determining unique combinations.

• Formulas for permutations and combinations are critical in combinatorial mathematics.

For more detailed study on this topic, you can refer to textbooks on discrete mathematics or combinatorics, such as "Discrete Mathematics and Its Applications" by Kenneth H. Rosen.