Question

In how many ways can 10 books on Mechanics and 8 books on Quantum Physics be placed in a row such that two Quantum Physics books are not together?

• A. 165
• B. 176
• C. 187
• D. 198

Hint:

Use the \textbf{gap method}: arrange the unrestricted group first, count the \(n+1\) gaps it creates, then choose gaps for the restricted items so that adjacency is impossible.

Answer 1

The Correct Option is A

Step 1: Clarify the model.
The answer choices are small, which indicates that books are considered \emph{identical within each subject} (only the subject matters). If every book were distinct, the count would be enormous and would not match the given options.
Step 2: Place the Mechanics books first to create safe slots.
Arrange the \(10\) Mechanics books in a row. This fixes the pattern \[ \_\,M\,\_\,M\,\_\,M\,\_\,\cdots\,M\,\_ \] There are \(10+1=11\) slots (gaps), including the two ends. To prevent two Quantum books from being adjacent, at most one Quantum book can be placed in any slot.
Step 3: Choose slots for the Quantum books.
We must choose \(8\) of the \(11\) slots to hold one Quantum book each. Since the Quantum books are identical and each chosen slot receives exactly one book, the count is simply \[ \binom{11}{8}=\binom{11}{3}=165. \] Step 4: No extra permutations.
There is no further multiplication: the Mechanics books are already fixed (identical), and within the chosen slots each receives exactly one identical Quantum book. Hence the number above is the final count.
\[ \boxed{165} \]