Question

A ‘To Do’ list contains 5 tasks unrelated to each other. All tasks have to be completedwithin 3 days. It is necessary to do at least 1 task everyday. How many different ways are there to make a schedule to complete these 5 tasks?

Generalize the result obtained in the previous exercise to n tasks and k days (n ≥ k ; there is at least one task scheduled for every day).

Answer 1

Let us first arrange the 5 tasks here as 5 sticks given as:

| | | | |

Now we can arrange 2 crosses here to divide them into 3 parts where each part would represent the task done on any particular day. The number of ways to do this is to select 2 crosses from the 4 crosses given as:

| X | X | X | X |
Therefore num of ways to choose: 4c2 = 6

But as all the tasks and days are non equivalent here, therefore we need to permutation of those tasks for each day in those 3 days.

Therefore total number of ways to do so is given as:
= 6*Number of permutations

= 6*3!

= 36

Therefore there are 36 ways to assign the 5 tasks in 3 days.

Now for n tasks and k days, we arrange the n tasks first. then we select (k - 1) crosses from (n - 1) crosses to divide them into k parts. And then multiply with k! to get the total number of ways. Therefore the general formula for the number of ways here is given as: