Permutations and Combinations
Gap Method
Premium Question
Grade 11
Question:
$m$ men and $n$ women ($m > n$) are to be seated in a row so that no two women are adjacent. The number of ways is:
(1) $m! \cdot \frac{(m + 1)!}{(m - n + 1)!}$
(2) $m! \cdot \frac{(m + 1)!}{(m + 1 - n)!}$
(3) $\frac{(m + 1)!}{(m - n)!}$
(4) $(m + 1)! \cdot ^mP_n$
Step-by-Step Solution
Key Concept: Seat $m$ men first: $m!$ ways creating $m + 1$ gaps. Choose $n$ of these gaps and place women in order: $P(m + 1, n) = \frac{(m+1)!}{(m+1-n)!}$. Total $= m! \cdot \frac{(m+1)!}{(m+1-n)!}$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)