A student can do the things bellow:

a. Do his homework in 2 days
b. Write a poem in 2 days
c. Go on a trip for 2 days
d. Study for exams for 1 day
e. Play pc games for 1 day
A schedule of n days can be completed by any combination of the activities above. For example 3 possible schedules for 7 days are:

homework, poem, homework, play
poem, study, play, homework, study
trip, trip, trip, study
Find a recursive function T(n) that represents the number of all possible schedules for n days.

I just need a start. Any ideas will be greatly appreciated.

Question

A student can do the things bellow:

a. Do his homework in 2 days
b. Write a poem in 2 days
c. Go on a trip for 2 days
d. Study for exams for 1 day
e. Play pc games for 1 day
A schedule of n days can be completed by any combination of the activities above. For example 3 possible schedules for 7 days are:

homework, poem, homework, play
poem, study, play, homework, study
trip, trip, trip, study
Find a recursive function T(n) that represents the number of all possible schedules for n days.

I just need a start. Any ideas will be greatly appreciated.

ganeshie8 · Answer

Basically you want to find the number of nonnegative integer solutions to below equation $$n=2a+2b+2c+d+e$$

yrelhan4 · Answer

So, T(n)=3T(n-2)+2T(n-1)?

ganeshie8 · Answer

may i know how you got that

yrelhan4 · Answer

Well to be honest, i don't exactly know how. 
I have been searching for examples over the net and this is how they write a recursive function.
For example, i read this question where there are n steps, and a person can either take 1 step at a time, or 2 steps at a time. We had to tell the number of ways to take n steps. and as you said, the equation that time would be a+2b=n.. and the recursive fucntion they wrote was t(n)=t(n-1)+t(n-2)..

so yeah I don't exactly know how. I have been taking this for granted. I'd be very happy if could tell me more about it.

ganeshie8 · Answer

check this http://math.stackexchange.com/questions/614356/difficult-recursion-problem

yrelhan4 · Answer

I checked that out before posting it here. Don't really understand it completely.

ganeshie8 · Answer

suppose $n=1$, how many ways can you schedule $1$ day ?

ganeshie8 · Answer

you can either `study` or `play`, so only $2$ ways, yes ?

ganeshie8 · Answer

that means 
$$T(1)=2$$

ganeshie8 · Answer

next let $n=2$ and work the number of ways $2$ days can be scheduled

yrelhan4 · Answer

I get it now. Thank you very much...
also, suppose i don't want the duplicates..
like, i treat "poem, study, play, homework, study" and "study, poem, play, homework, study" the same...
how do I draft an algo for that?
@ganeshie8

yrelhan4 · Answer

@Luigi0210

yrelhan4 · Answer

@Nurali

anonymous · Answer

no idea how you can even solve this its more of an opinion than a math question.