Question

An identification code read from left to right consists of 2 digits, a dash, 3 digits, a dash, and then 4 digits. Each digit can be any number from 0 through 9.
Column A: The number of different identification codes possible
Column B: 10⁹

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

For problems involving sequences of choices (like digits in a code, letters in a password, etc.), the fundamental counting principle is the key. Multiply the number of options for each position to get the total number of possibilities.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question is about counting the total number of possible combinations for an identification code based on a specific structure, which is a core topic in combinatorics.
Step 2: Key Formula or Approach:
The fundamental counting principle states that if there are \(n_1\) ways for the first event to occur, \(n_2\) ways for the second, ..., and \(n_k\) ways for the \(k\)-th event, then the total number of ways for the sequence of events to occur is \(n_1 \times n_2 \times \dots \times n_k\).
Step 3: Detailed Explanation:
Column A: We need to calculate the total number of possible identification codes.
The format of the code is DD-DDD-DDDD, where D represents a digit.
Each digit can be any number from 0 through 9. This means there are 10 possible choices for each digit position (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
The total number of digits in the code is \(2 + 3 + 4 = 9\).
For each of these 9 positions, there are 10 independent choices.
Using the fundamental counting principle, the total number of different codes is:
\[ \underbrace{10 \times 10}_{\text{2 digits}} \times \underbrace{10 \times 10 \times 10}_{\text{3 digits}} \times \underbrace{10 \times 10 \times 10 \times 10}_{\text{4 digits}} = 10^2 \times 10^3 \times 10^4 \] Using the rule of exponents (\(x^a \times x^b = x^{a+b}\)):
\[ 10^{2+3+4} = 10^9 \] So, the quantity in Column A is \(10^9\).
Column B: The quantity is given as \(10^9\).
Comparison:
The quantity in Column A is \(10^9\), and the quantity in Column B is \(10^9\). The two quantities are equal.
Step 4: Final Answer:
The total number of possible codes is \(10^9\), which is equal to the quantity in Column B.