<p>Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place is \(l\), then the sum of digits of \(l\) is?</p>
Step-by-Step Solution
Key Concept: Use the inclusion-exclusion principle to count committees where no couple is selected together. We subtract cases where at least one couple is together from the total number of committees.
<p><strong>Step 1: Find total committees of 4 from 8 people.</strong></p><p>Total ways to select 4 members from 4 couples (8 people) = C(8,4) = 70</p><p><strong>Step 2: Apply inclusion-exclusion principle.</strong></p><p>Let A_i = committees containing the i-th couple (i = 1,2,3,4)</p><p>We need: Total - |A₁ ∪ A₂ ∪ A₃ ∪ A₄|</p><p><strong>Step 3: Calculate |A_i| (committees containing at least one specific couple).</strong></p><p>If couple i is in the committee, we need 2 more members from remaining 6 people.</p><p>|A_i| = C(6,2) = 15</p><p>Sum of individual sets: |A₁| + |A₂| + |A₃| + |A₄| = 4 × 15 = 60</p><p><strong>Step 4: Calculate |A_i ∩ A_j| (committees with at least two specific couples).</strong></p><p>If two couples are in the committee, all 4 seats are filled.</p><p>|A_i ∩ A_j| = C(4,0) = 1</p><p>Number of pairs: C(4,2) = 6</p><p>Sum: 6 × 1 = 6</p><p><strong>Step 5: Calculate |A_i ∩ A_j ∩ A_k| and higher intersections.</strong></p><p>If three couples (6 people) are selected, we can only choose 4, which is impossible without leaving someone out. So these intersections are 0.</p><p><strong>Step 6: Apply inclusion-exclusion formula.</strong></p><p>|A₁ ∪ A₂ ∪ A₃ ∪ A₄| = 60 - 6 + 0 - 0 = 54</p><p><strong>Step 7: Find committees with no couples.</strong></p><p>l = 70 - 54 = 16</p><p><strong>Step 8: Find sum of digits.</strong></p><p>l = 16</p><p>Sum of digits = 1 + 6 = 7</p><p><strong>∴ Answer: 7</strong></p>
Correct Answer: 7