Question
Three (3) different science books and four (4) different MAPEH books are to be arranged on a shelf. How many possible arrangements can be made if the books are arranged alternately?
Answers
Bot
First, we need to choose which type of book goes first, science or MAPEH. There are two possibilities.
If we start with a science book, there are 3 ways to choose which book goes first, then 4 ways to choose which MAPEH book goes next, then 2 ways to choose which science book goes next, and so on. This gives a total of:
3 x 4 x 2 x 3 x 1 x 2 x 1 = 144
possible arrangements.
If we start with a MAPEH book, there are 4 ways to choose which book goes first, then 3 ways to choose which science book goes next, then 3 ways to choose which MAPEH book goes next, and so on. This gives a total of:
4 x 3 x 3 x 2 x 2 x 1 x 1 = 144
possible arrangements.
Therefore, the total number of possible arrangements if the books are arranged alternately is:
144 + 144 = 288
If we start with a science book, there are 3 ways to choose which book goes first, then 4 ways to choose which MAPEH book goes next, then 2 ways to choose which science book goes next, and so on. This gives a total of:
3 x 4 x 2 x 3 x 1 x 2 x 1 = 144
possible arrangements.
If we start with a MAPEH book, there are 4 ways to choose which book goes first, then 3 ways to choose which science book goes next, then 3 ways to choose which MAPEH book goes next, and so on. This gives a total of:
4 x 3 x 3 x 2 x 2 x 1 x 1 = 144
possible arrangements.
Therefore, the total number of possible arrangements if the books are arranged alternately is:
144 + 144 = 288
Tammy
If they must be arranged alternately, we cannot start with the science books,
because we would have 2 MAPEH books side by side at the other end, so
we must start with a MAPEH book, e.g. SMSMSMM
arrangement: MSMSMSM
number of arrangements = 4*3*3*2*2*1*1 = 144
because we would have 2 MAPEH books side by side at the other end, so
we must start with a MAPEH book, e.g. SMSMSMM
arrangement: MSMSMSM
number of arrangements = 4*3*3*2*2*1*1 = 144
Bot
You are correct. I apologize for my mistake. Thank you for pointing it out and providing the correct solution. The number of possible arrangements if they must be arranged alternately, starting with a MAPEH book, is indeed 4 x 3 x 3 x 2 x 2 x 1 x 1 = 144.