There are 6 periods in each working day of a school. In how many ways can one organize 5 subjects such that each subject is allowed at least one period?
There are 5! = 120 ways to arrange 5 subjects.
Now, the 6th subject (which just happens to be one of the 5 as well) can be inserted in any place in the schedule. So, there are 6 places to insert it. The 6th subject can be any of the 5, so there are 5 ways to choose it.
So, there are 5! * 6 * 5 ways to arrange the subjects, but some arrangements will have the same subject for two periods in a row.
To solve this problem, we can use the concept of distributing identical objects into distinct groups. Let's break down the problem:
We have 5 subjects to organize, and each subject must be allocated at least one period. This means that we have 5 objects (subjects) and 6 distinct groups (periods).
Step 1: Assign one subject to each of the 6 periods to ensure that all subjects have at least one period.
Since we have 5 subjects and 6 periods, we can assign one subject to each period in 5! (read as "5 factorial") ways. Factorial is denoted by the exclamation mark (!), and it represents the product of all positive integers less than or equal to a given positive integer.
So far, we have 5! = 5 x 4 x 3 x 2 x 1 = 120 ways to allocate one subject to each period.
Step 2: Distribute the remaining 4 subjects among the 6 periods.
To distribute the remaining 4 subjects among the 6 periods, we can use a technique called stars and bars. This technique involves counting the number of ways to distribute identical items among distinct groups.
In this case, we can represent the 4 remaining subjects as stars (*) and represent the 6 periods as bars (|). For example, a possible distribution could be: * | * * | * * | *.
The total number of ways to distribute the 4 subjects among the 6 periods can be calculated using the formula (n + r - 1) choose (r - 1). In this formula, n represents the number of subjects to distribute, and r represents the number of groups.
Therefore, the number of ways to distribute the 4 subjects among the 6 periods is (4 + 6 - 1) choose (6 - 1) = 9 choose 5 = 126.
Step 3: Multiply the results from Step 1 and Step 2 to get the final answer.
To obtain the total number of ways to organize the 5 subjects, multiply the number of ways to assign one subject to each period (Step 1) by the number of ways to distribute the remaining subjects (Step 2).
Final Answer: 120 x 126 = 15,120
Therefore, there are 15,120 ways to organize the 5 subjects such that each subject is allowed at least one period.