The digits $1,2,3,4$ and $5$ can be arranged to form many different $5$-digit positive integers with five distinct digits. In how many such integers is the digit $1$ to the left of the digit $2$?  (The digits 1 and 2 do not have to be next to each other.)
For the first digit, we have 5 choices, then we have 4 choices left for the second digit, then 3 choices for the third digit, etc. So there are $5!=120$ arrangements of the digits. Notice that for each arrangement with 1 to the left of 2, we can reverse the arrangement so that 2 is to the left of 1. For instance, flipping 31245 results in 54213. So by symmetry, exactly half of the arrangements have 1 to the left of 2. In $\frac{120}{2}=\boxed{60}$ integers, the digit 1 is to the left of the digit 2.
60
Level 4
Counting & Probability
Problem: The digits $1,2,3,4$ and $5$ can be arranged to form many different $5$-digit positive integers with five distinct digits. In how many such integers is the digit $1$ to the left of the digit $2$?  (The digits 1 and 2 do not have to be next to each other.)
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the number of 5-digit integers formed by the digits 1, 2, 3, 4, and 5 with the condition that digit 1 is to the left of digit 2, we first calculate the total number of arrangements of the five distinct digits. The total arrangements can be calculated using the factorial of the number of digits, which is 5! = 120.

Next, we recognize that in any arrangement of the digits, the digit 1 can either be to the left of digit 2 or to the right of digit 2. These two scenarios are mutually exclusive and collectively exhaustive. Therefore, the arrangements where digit 1 is to the left of digit 2 will be exactly half of the total arrangements.

Thus, we divide the total arrangements by 2 to find the number of arrangements where digit 1 is to the left of digit 2:

Number of arrangements with 1 to the left of 2 = Total arrangements / 2 = 120 / 2 = 60.

This leads us to conclude that the final answer is 60.<|END_OF_LATENT|> The final answer is $60$.