Permutations & Combinations
Arrangements with restrictions
Grade 11
Question:
<p>There are six teachers. Out of them two are primary teachers, two are middle teachers, and two are secondary teachers. They are to stand in a row, so as the primary teachers, middle teachers, and secondary teachers are always in a set. Find the number of ways in which they can do so.</p>
Step-by-Step Solution
Key Concept: Treat each category (primary, middle, secondary) as a single unit first, then arrange units and arrange teachers within each unit separately using the multiplication principle.
<p><strong>Step 1:</strong> Identify the constraint. Teachers must stand in groups by their category (primary, middle, secondary). Treat each category as one unit/block.</p><p><strong>Step 2:</strong> Arrange the 3 blocks (primary block, middle block, secondary block) in a row. Number of ways = 3! = 6</p><p><strong>Step 3:</strong> Within the primary block, 2 primary teachers can arrange themselves in 2! = 2 ways.</p><p><strong>Step 4:</strong> Within the middle block, 2 middle teachers can arrange themselves in 2! = 2 ways.</p><p><strong>Step 5:</strong> Within the secondary block, 2 secondary teachers can arrange themselves in 2! = 2 ways.</p><p><strong>Step 6:</strong> By multiplication principle, total arrangements = 3! × 2! × 2! × 2! = 6 × 2 × 2 × 2 = 6 × 8 = 48</p><p>∴ Answer: <strong>48</strong></p>
Correct Answer: 48