<p>A boat is to be manned by 9 crew with 4 on the stroke side, 4 on the row side and one to steer. There are 11 crew of which 2 can stroke only, 1 can row only while 3 can steer only. In how many ways the crew can be arranged for the boat?</p>
Step-by-Step Solution
Key Concept: Partition the 11 crew into three constraint groups: 2 stroke-only, 1 row-only, 3 steer-only, and 5 universal crew. The steer position must use one of the 3 steer-only members, then select and arrange the remaining 8 positions from the remaining 10 crew, respecting stroke/row constraints.
<p><strong>Step 1: Fix the Steersman</strong> Choose 1 from 3 steer-only crew: 3 ways</p><p><strong>Step 2: Analyze Remaining Crew</strong> After fixing steersman, remaining crew: 2 stroke-only, 1 row-only, 3 steer-only (unused), 5 universal = 10 available for 8 positions</p><p><strong>Step 3: Fill Stroke Side (4 positions)</strong> Must use both 2 stroke-only members + choose 2 from 5 universal crew: C(5,2) = 10 ways. Arrange 4 on stroke side: 4! = 24 ways. Subtotal: 10 × 24 = 240</p><p><strong>Step 4: Fill Row Side (4 positions)</strong> Must use 1 row-only member + choose 3 from remaining 3 universal crew: C(3,3) = 1 way. Arrange 4 on row side: 4! = 24 ways. Subtotal: 1 × 24 = 24</p><p><strong>Step 5: Calculate Total</strong> Total = 3 (steersman) × 240 (stroke side) × 24 (row side) = 3 × 240 × 24 = 17,280</p><p>∴ <strong>Answer: 17,280</strong></p>
Correct Answer: 17