Question

i have $7$ books i want to arrange on a shelf. two of them are math books, and one is a physics book. how many ways are there for me to arrange the books if i want to put the math books next to each other, and put both of them to the left of the physics book? (note that all the books are distinguishable.)

Answer 1

To arrange 7 books with 2 math books together and to the left of a physics book, there are 240 different ways to do so.

Step-by-step explanation:

The question involves arranging 7 books on a shelf with the constraint that 2 math books must be together and to the left of 1 physics book. To solve this, we first treat the 2 math books as one unit since they must be together. Together with the 1 physics book, we effectively have 6 items to arrange (the unit of math books, 4 other books, and the physics book).

There are 5! (factorial) ways to arrange these 6 items. However, within the math books unit, there are 2! ways to arrange the two books with respect to each other. Our arrangements must also ensure the math unit is to the left of the physics book. To ensure this, position the physics book and consider the remaining spots to the left for the other items.

With the physics book in any of the 5 positions from the left, we have 4 remaining positions before it. There are 4! ways to arrange the 4 non-math and non-physics books in these slots. Therefore, the total number of arrangements is 5 * 4! * 2!.

Calculating this gives us: 5 * (4 * 3 * 2 * 1) * (2 * 1) = 5 * 24 * 2 = 240 ways to arrange the books while satisfying all constraints.