Five math books and three physics books are being placed on a shelf. How many ways can the books be arranged so that the physics books are next to each other?
The correct answer and explanation is:
To solve the problem of arranging five math books and three physics books on a shelf such that the physics books are next to each other, we can treat the three physics books as a single unit or “block” because they must be adjacent.
Step 1: Consider the physics books as a block
By grouping the three physics books together as a block, we now have five math books and one physics book block. This reduces the total number of units to be arranged to six (five math books + one block of physics books).
Step 2: Arrange the six units
The total number of ways to arrange these six units is simply the number of permutations of six objects, which is 6 factorial (6!). The formula for this is: 6!=6×5×4×3×2×1=7206! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
Step 3: Arrange the physics books within the block
Now, within the physics book block, there are three physics books that can be arranged among themselves. The number of ways to arrange the three physics books within the block is 3 factorial (3!), which is: 3!=3×2×1=63! = 3 \times 2 \times 1 = 6
Step 4: Multiply the results
To find the total number of ways to arrange the books, we multiply the number of ways to arrange the six units (which includes the physics book block) by the number of ways to arrange the books within the physics book block. This gives: 6!×3!=720×6=43206! \times 3! = 720 \times 6 = 4320
Thus, the total number of ways to arrange the five math books and three physics books on the shelf such that the physics books are next to each other is 4,320.
This solution method ensures that the physics books remain together while calculating all possible arrangements of the books.