Which is the total number of arrangements of the digits 1, 2, 3, 4, 5, if the even digits must not be together?
To find the total number of arrangements where the even digits are not together, we can subtract the number of arrangements where the even digits are together from the total number of arrangements.
The total number of arrangements of the digits 1, 2, 3, 4, 5 is 5! = 5 x 4 x 3 x 2 x 1 = 120.
Now let's count the number of arrangements where the even digits (2 and 4) are together. We can treat 2 and 4 as a single entity, so we have a total of 4 entities: {24, 1, 3, 5}, {1, 24, 3, 5}, {1, 3, 24, 5}, {1, 3, 5, 24}. For each arrangement of the 4 entities, there are 2 ways to arrange the even digits within the {2, 4} entity. So the total number of arrangements where the even digits are together is 4 x 2 = 8.
Finally, we subtract the number of arrangements where the even digits are together from the total number of arrangements: 120 - 8 = <<120-8=112>>112.
Therefore, the total number of arrangements of the digits 1, 2, 3, 4, 5, if the even digits must not be together, is 112.
To find the total number of arrangements of the digits 1, 2, 3, 4, 5, where the even digits must not be together, we can use the principle of inclusion-exclusion.
Step 1: Calculate the total number of arrangements without any restrictions.
The total number of arrangements of 5 digits is 5! (factorial), which is equal to 5 x 4 x 3 x 2 x 1 = 120.
Step 2: Calculate the number of arrangements when the even digits (2 and 4) are together.
Treat the 2 even digits as a single unit. This reduces the problem to arranging 4 units (1, 3, 5, and the even digit unit) instead of 5.
The number of arrangements of these 4 units is 4! = 4 x 3 x 2 x 1 = 24.
However, within the even digit unit, there are 2 ways to arrange the even digits. Since there are two even digits, we multiply the number of arrangements by 2.
So, the number of arrangements when the even digits are together is 24 x 2 = 48.
Step 3: Calculate the number of arrangements when both 2 and 4 are not together.
Since we already calculated the number of arrangements when the even digits were together, we subtract that from the total number of arrangements calculated in Step 1.
Total arrangements - Arrangements when even digits are together = 120 - 48 = 72.
Therefore, the total number of arrangements of the digits 1, 2, 3, 4, 5, where the even digits must not be together, is 72.
To find the total number of arrangements of the digits 1, 2, 3, 4, 5, where the even digits (2 and 4) must not be together, we need to consider all possible arrangements and subtract the ones that have the even digits together.
Step 1: Total number of arrangements
Since we have 5 digits (1, 2, 3, 4, 5), and we want to arrange them without any restrictions initially, we have all 5 digits available for the first place, 4 left for the second place, 3 for the third, 2 for the fourth, and 1 for the fifth place. This can be calculated using the factorial function denoted by "!", which represents the product of all positive integers less than or equal to a given number.
Total number of arrangements = 5! = 5 * 4 * 3 * 2 * 1 = 120.
So, there are 120 total arrangements without any restrictions.
Step 2: Calculate the number of arrangements with the even digits together
To calculate the number of arrangements where the even digits (2 and 4) are together, we can treat them as a single entity (24) and find the number of arrangements for this entity and the other three digits (1, 3, 5). Since there are 4 digits (24, 1, 3, 5), we can arrange them among themselves in 4! ways.
Number of arrangements with even digits together = 4! = 4 * 3 * 2 * 1 = 24.
Step 3: Calculate the number of arrangements with even digits not together
To find the number of arrangements where the even digits are not together, we subtract the number of arrangements with the even digits together from the total number of arrangements.
Number of arrangements with even digits not together = Total number of arrangements - Number of arrangements with even digits together
Number of arrangements with even digits not together = 120 - 24 = 96.
Therefore, the total number of arrangements of the digits 1, 2, 3, 4, 5, where the even digits must not be together, is 96.