Problem 31 - Algebra 2 and Trigonometry

Question

From the set {2, 3, 4, 5, 6}, a number is drawn and replaced. Then a second number is drawn. How many two-digit numbers can be formed?

Step-by-Step Solution

Answer

There are 25 two-digit numbers that can be formed.

Step 1: Identify Possible Outcomes for Each Draw

From the set \( \{2, 3, 4, 5, 6\} \), there are 5 possible numbers that can be drawn in each draw because the number is replaced each time.

Step 2: Determine Total Number of Two-Digit Combinations

Since each digit of the two-digit number can be any of the 5 numbers in the set, the total two-digit combinations can be calculated by multiplying the possibilities for the first digit by the possibilities for the second digit. Thus, there are \(5 \times 5 = 25\) possible two-digit numbers.

Key Concepts

Two-Digit Numbers

Two-digit numbers are numbers that have two digits, typically ranging from 10 to 99 in standard numeric systems. However, in this scenario, we are creatively generating two-digit numbers by picking two numbers successively from the specified set \( \{2, 3, 4, 5, 6\} \). This means that each digit of the number is independently chosen from these five options. The first digit is the tens digit, and the second digit is the units digit. Remember that since these numbers do not need to start from ten, any combination of two numbers from the set can form a new two-digit number.

Replacement in Probability

The concept of replacement in probability is essential when drawing items from a set. When a number is drawn and replaced, it means that after choosing the first number, it is put back into the set. This allows for the possibility of drawing the same number again in a subsequent draw. In our exercise, after choosing a number from the set \( \{2, 3, 4, 5, 6\} \), it is replaced. This results in each draw being independent, maintaining the same number of choices for each digit of the two-digit number. Replacement ensures that the total number of outcomes remains consistent across draws.

Possible Outcomes

In probability, the term 'possible outcomes' refers to all the possible results that can occur. When drawing from the set \( \{2, 3, 4, 5, 6\} \) twice, each draw independently offers 5 options. These options create the range of possibilities for each digit of the two-digit number. Since each draw is independent due to replacement, for the first digit, there are 5 outcomes; similarly, for the second digit, there are also 5 outcomes. These combined allow for a multitude of combinations, all contributing to form the total variation of possible two-digit numbers.

Multiplicative Principle

The multiplicative principle is a core concept in combinatorics. It states that if one event can occur in \( m \) ways, and a second event can occur independently in \( n \) ways, then the two events together can occur in \( m \times n \) ways. This principle is applied to find the total number of two-digit numbers in this scenario. Here, both the tens and units digits of the number can be any of 5 numbers from the set \( \{2, 3, 4, 5, 6\} \). Thus, using the multiplicative principle, the number of possible two-digit numbers is calculated as \( 5 \times 5 = 25 \). This multiplication captures all possible pairings of digits from the available options.