- Kate is planning for her $8$ days Study Period.
- Each day she can choose one of the $3$ Subjects: Math, English or
Physics. - She never studies Math and English on consecutive days. (i.e.No ME or EM)
- She also wants to study at least all $3$ subjects on at least one day of
her study period.
How many different schedules are possible?.
I tried $3^4-3 \cdot 2^4+3 \cdot 1= 36$ for $4$ day schedule. I draw the picture and there shall be only $10$ possible schedules. Not sure how to do exclusion on no math and English on consecutive days. Please help. Thank you.
Best Answer
In Day1, if P chosen, then all 3 (P, M, E) on Day 2. If she chose M, then she only has 2 (P,M) on day 2 and if she chose E she also only has 2 (P, E) for day 2. Add up all the possibilities she has total 7 cases for day 2.
In day 3 and in all of the scenarios on the day 2, she can do P, so she has 3+2+2=7 options. If she chose E she has 3+2=5 and the same for M. Adds all possibility for Day 3 is 17.
M 1 2 5 12 29 70 169 408
E 1 2 5 12 29 70 169 408
P 1 3 7 17 41 99 239 577
Total3 7 17 41 99 239 577 1393
Take Away not all 3 subjects in the schedule = 2*2^8(number of days)= 512
Add 1 for undercounting
882