A security alarm requires a four-digit code. The code can use the digits 0–9 and the digits cannot be repeated.
Which expression can be used to determine the probability of the alarm code beginning with a number greater than 7?
The Correct Answer and Explanation is:
Correct Answer: 2×9×8×710×9×8×7\frac{2 \times 9 \times 8 \times 7}{10 \times 9 \times 8 \times 7}
Explanation
We are given a four-digit security alarm code that uses digits from 0 to 9 without repetition, and we are asked to find the probability that the code begins with a number greater than 7.
Step 1: Total number of possible 4-digit codes (no repetition)
The total number of 4-digit codes using digits 0–9 without repetition is:
- First digit: 10 options (0–9)
- Second digit: 9 options (one digit used already)
- Third digit: 8 options
- Fourth digit: 7 options
So, the total number of possible codes is: 10×9×8×710 \times 9 \times 8 \times 7
Step 2: Favorable outcomes — codes starting with a digit > 7
Digits greater than 7 are 8 and 9 — that gives us 2 choices for the first digit.
Once the first digit is selected (either 8 or 9), we have:
- Second digit: 9 choices (excluding the first digit)
- Third digit: 8 choices (excluding the first two digits)
- Fourth digit: 7 choices
So, favorable outcomes: 2×9×8×72 \times 9 \times 8 \times 7
Step 3: Compute probability
Probability = Favorable outcomes ÷ Total outcomes Probability=2×9×8×710×9×8×7\text{Probability} = \frac{2 \times 9 \times 8 \times 7}{10 \times 9 \times 8 \times 7}
Notice that the common terms (9 × 8 × 7) cancel out: 210=0.2\frac{2}{10} = 0.2
Thus, the probability that a randomly selected 4-digit code with no repeated digits begins with a digit greater than 7 is 0.2, or 20%.
The given expression correctly models this probability and highlights the importance of identifying constraints (no repetition) and specific conditions (first digit > 7) in probability problems.
