Understanding Divisibility by 4
To determine if a number is divisible by 4, we only need to consider the last two digits of that number. Therefore, we will focus on the combinations of the last two digits formed from the selection of the digits 1, 2, 5, 6, and 7.
Possible Combinations of Last Two Digits
The last two digits must form a number that is divisible by 4. The possible two-digit combinations from the digits provided are:
- 12
- 16
- 25
- 26
- 32
- 52
- 56
- 72
- 76
From these, the valid combinations that are divisible by 4 are:
- 12 (12 ÷ 4 = 3)
- 16 (16 ÷ 4 = 4)
- 56 (56 ÷ 4 = 14)
Total Combinations for Last Two Digits
- Total valid combinations = 3
Choosing the First Two Digits
Once we select the last two digits, we can choose any two of the remaining digits for the first two positions. Since we have 5 digits total and use 2 for the last positions, we have 3 digits left.
The number of ways to choose 2 digits from 3 is given by:
- 3! = 6 ways
Total Valid 4-Digit Numbers
For each valid pair of last digits, we can arrange the first two digits in 2! = 2 ways. Thus, the total valid combinations are:
- Total = 3 (last two combinations) * 6 (first two combinations) * 2 (arrangements)
- Total = 36 valid 4-digit numbers
Calculating Total Possible 4-Digit Combinations
The total number of 4-digit combinations that can be formed from the 5 digits (1, 2, 5, 6, 7) without repetition is calculated as follows:
- Total = 5P4 = 120 (since we can select 4 digits out of 5)
Finding the Probability
Finally, the probability that a randomly formed 4-digit number is divisible by 4 is:
- Probability = Total valid 4-digit numbers / Total possible 4-digit combinations
- Probability = 36 / 120 = 3 / 10
Thus, the probability that a randomly formed 4-digit number using the digits 1, 2, 5, 6, and 7 is divisible by 4 is 3/10.