<p>In how many ways can a committee of 10 be selected with at least 4 women and 4 men from 9 women and 8 men if<br>(i) Ms X refuses to work with Mr Y<br>(ii) Ms X and Mr Y insist to work together?</p>
Step-by-Step Solution
Key Concept: Handle the constraint by calculating total valid committees minus the forbidden case (for part i), and for part ii, treat Ms X and Mr Y as a single unit, then adjust the remaining selections accordingly.
<p><strong>Part (i): Ms X refuses to work with Mr Y</strong></p><p><strong>Step 1:</strong> Find total committees with at least 4 women and 4 men from 9 women and 8 men.</p><p>Valid distributions: (4W,6M), (5W,5M), (6W,4M)</p><p>Total = C(9,4)·C(8,6) + C(9,5)·C(8,5) + C(9,6)·C(8,4)</p><p>= 126×28 + 126×56 + 84×70 = 3528 + 7056 + 5880 = 16464</p><p><strong>Step 2:</strong> Subtract committees with both Ms X and Mr Y.</p><p>When both are included, select 3 more women from 8 and 3 more men from 7:</p><p>Invalid = C(8,3)·C(7,3) + C(8,4)·C(7,2) + C(8,5)·C(7,1)</p><p>= 56×35 + 70×21 + 56×7 = 1960 + 1470 + 392 = 3822</p><p><strong>Step 3:</strong> Answer for (i) = 16464 - 3822 = 12642</p><p><strong>Part (ii): Ms X and Mr Y insist to work together</strong></p><p><strong>Step 4:</strong> Treat Ms X and Mr Y as one unit. Now select 8 more from remaining 16 people (8W and 7M) with at least 3 more women and 3 more men.</p><p>Valid distributions: (3W,5M), (4W,4M), (5W,3M)</p><p>Total = C(8,3)·C(7,5) + C(8,4)·C(7,4) + C(8,5)·C(7,3)</p><p>= 56×21 + 70×35 + 56×35 = 1176 + 2450 + 1960 = 5586</p><p>∴ Answer for (ii): 5586</p><p><em>Note: The given answer 16464 represents the total valid committees before applying constraint (i).</em>
Correct Answer: 16464