How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

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How many ways can you arrange 4 physics book and 5 math books so that the physics book are next to each other?

So I know the arrangements can be,

PPPPMMMMM MPPPPMMMM MMPPPPMMM MMMPPPPMM

MMMMMPPPP

4! * 5! = 2880 ways
There are 4 ways to arrange the physics book and 5 ways to arrange the others so 4! * 5!. I'm not sure if that's the correct answer.

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A. There are 9 distinguishable items (books), so they can be arranged in 9! ways. (9 choices for the first book, times 8 for the 2nd book, �, times 2 for the 8th book, times 1 for the 9th book). � B. Keeping the same subjects together, there are 4! ways to arrange the mathematics books, and 5! ways to arrange the science books. Additionally, you can have the groups themselves arranged two ways (one with math on the left, the other with math books on the right): 4!*5!*2 � C. For this arrangement, the science books need to be on the ends (S-M-S-M-S-M-S-M-S) in order to 'sandwich' the math books. There are 5 places to hold science books and 4 places for math books, so I'd say 5!*4! ways to arrange all the books like this. �

I think this is correct, but another tutor's opinion/verification might be a good idea.

Answer:

a. 362,880

b. 5,760

c. 2,880

Step-by-step explanation:

For this problem, let us discuss the concept of permutation. A permutation is a act of arranging things into a sequence. Sometimes, these things are taken all together, sometimes, only parts of it are arranged. The keyword for permutation is order. Order is vital to counting arrangement when it comes to permutation.

The formula for arranging r things from n possible ones, where order matters is given by the formula:

How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

The "!" symbol is called "factorial". In simple terms,

x! = (x)(x-1)(x-2)...(3)(2)(1).

For example, 5! is 5*4*3*2*1 = 120. Also, a factorial is expressed as a product, so you can cancel out factors of factorials when they are present in a fraction.

note that 0! = 1

If we are arranging n things in a row, and you use all of them, the formula is reduced to.

How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

In general, there are n! ways of arranging n things in a row.

a. There are no restriction?

Since there 9 total books, and you have to arrange all 9 of them, we have n = 9, and r = 9.

Substituting to the formula gives us

How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

There are 362,880 ways to arrange the 9 different books.

b. Books of the same subject must be placed together?

We can treat this problem as arranging the 2 subjects first in a row. The 2 subjects are math and science. If we arrange 2 objects in a row, our formula uses n=2, r=2.

How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

There are 2 ways to arrange each subjects.

Now let us look into detail each subjects. There are 4 math books, and 5 science books. The 4 math books can arrange themselves as long as they are together. The 5 science books can also arrange themselves as long as they are still together.

For the math books, there are 4 books and we arrange all 4 of them. Earlier, we said that if we arrange n things in a row, there are n! ways of arranging them.

How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

There are 24 ways to arrange the 4 math books in a row.

The same can be said for the 5 science books.

How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

There are 120 ways of arranging the 5 science books in a row.

From the fundamental principles of counting, product rule, we know that:

If there are x ways of doing something, and y ways of doing something else, then there are x*y ways of doing both actions.

Here are our actions:

  • Arranging the categories (science and math) = 2 ways
  • Arranging the 4 math books = 24 ways
  • Arranging the 5 science books = 120 ways

We want all of those to happen so we multiply all of them together.

How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

There are 5,760 ways of arranging the books if each subject must be placed together.

c. If they must be placed alternately?

In this problem, it is easier to represent each book with a blank (_) and put numbers on it based on how many books we can place. We then multiply all the results since we want all of them to happen (product rule).

_ _ _ _ _ _ _ _ _

There are 5 science books that can be placed on the 1st blank.

5 _ _ _ _ _ _ _ _

The next book should be a math book since they should alternate.

5 4 _ _ _ _ _ _ _

The next book should be a science book, but there are 4 of them left.

5 4 4 _ _ _ _ _ _

This pattern goes on to complete all the blanks.

5 4 4 3 3 2 2 1 1

We multiply these numbers.

How many ways can 4 different math books and 5 different science books be arranged on a shelf of books of the same subject are together?

(Note: We cannot have a scenario with math books going first since they are fewer than the science books. Putting math books first makes 2 science books go last, which does not satisfy the problem)

There are 2,880 ways of arranging the books if the subjects alternate.

For more information about permutations, click here.

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