The digits 1,2,3,4,5,6 are arranged to form a 6-digit number.What is the probability that the number formed is divisible by 4?
The last two digits will need to be divisible by 4.
That means we may have
12,16
24
32,36
44
52,56
64
Or, 9 possible 2-digit endings
Since there are 6*5 = 30 total endings, there is a 30% chance that the number is divisible by 4.
To find the probability that the number formed is divisible by 4, we need to determine the total number of arrangements that are divisible by 4 and divide it by the total number of possible arrangements.
First, let's determine the total number of possible arrangements. Since there are 6 digits to arrange (1, 2, 3, 4, 5, 6), there are 6 choices for the first digit, 5 choices for the second digit (since we cannot repeat the first digit), 4 choices for the third digit, and so on until we have only 1 choice for the sixth digit.
Therefore, the total number of possible arrangements is: 6 × 5 × 4 × 3 × 2 × 1 = 720.
Now, let's determine the number of arrangements that are divisible by 4. To be divisible by 4, the last two digits of the number must form a number that is divisible by 4.
There are three cases to consider:
Case 1: The last two digits are 12. In this case, there are 4 choices for the remaining four digits (3, 4, 5, 6), since we cannot repeat any of the digits. So, there are 4 × 1 × 1 = 4 arrangements.
Case 2: The last two digits are 24. Again, there are 4 choices for the remaining four digits. So, there are 4 × 1 × 1 = 4 arrangements.
Case 3: The last two digits are 32. There are 4 choices for the remaining four digits in this case as well. So, there are 4 × 1 × 1 = 4 arrangements.
Therefore, the total number of arrangements that are divisible by 4 is: 4 + 4 + 4 = 12.
Finally, we can calculate the probability by dividing the number of arrangements that are divisible by 4 by the total number of possible arrangements:
Probability = (Number of arrangements divisible by 4) / (Total number of possible arrangements)
= 12 / 720
= 1 / 60
So, the probability that the number formed is divisible by 4 is 1/60.