In how many ways can 4 books be arranged on a shelf if 2 specific books must be

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Question: In how many ways can 4 books be arranged on a shelf if 2 specific books must be together?

Options:

Correct Answer: 48

Solution:

Treat the 2 specific books as one unit. Then, we have 3 units to arrange: (2 books together) + (2 other books) = 3! * 2! = 12.

In how many ways can 4 books be arranged on a shelf if 2 specific books must be together?

In how many ways can 4 books be arranged on a shelf if 2 specific books must be together?

Step 1: Identify the 2 specific books that must be together. Let's call them Book A and Book B.
Step 2: Treat Book A and Book B as one single unit or 'block'. Now, instead of 4 separate books, we have 3 units to arrange: the 'block' (Book A and Book B together) and the other 2 books (Book C and Book D).
Step 3: Count the number of units we have. We have 3 units: (Book A and Book B together), Book C, and Book D.
Step 4: Calculate the number of ways to arrange these 3 units. The number of arrangements of 3 units is given by 3! (3 factorial), which is 3 x 2 x 1 = 6.
Step 5: Now, within the 'block' of Book A and Book B, these 2 books can also be arranged in different ways. The number of arrangements of these 2 books is given by 2! (2 factorial), which is 2 x 1 = 2.
Step 6: To find the total number of arrangements, multiply the number of arrangements of the 3 units by the arrangements of the 2 books in the block: 3! * 2! = 6 * 2 = 12.

Permutations – The arrangement of items in a specific order, where the order matters.
Grouping – Treating a set of items as a single unit to simplify the arrangement process.