<p>A delegation of four students is to be selected from a total of 12 students. In how many ways can the delegation be selected in each of the following cases?</p><p>(i) if all the students are equally willing?</p><p>(ii) if two particular students have to be included in the delegation?</p><p>(iii) if two particular students do not wish to be together in the delegation?</p><p>(iv) if two particular students wish to be included together only in the delegation?</p><p>(v) if two particular students refuse to be together and two other particular students wish to be together only in the delegation?</p>
Step-by-Step Solution
Key Concept: Break down complex selection constraints by identifying which students must be included, excluded, or have conditional restrictions—then apply combinations systematically to each case rather than trying to handle all constraints simultaneously.
<p><strong>(i) All students equally willing:</strong></p><p>Select 4 from 12 without restrictions: C(12,4) = 12!/(4!×8!) = (12×11×10×9)/(4×3×2×1) = <strong>495</strong></p><p><strong>(ii) Two particular students must be included:</strong></p><p>Fix 2 students, select remaining 2 from 10: C(10,2) = (10×9)/2 = <strong>45</strong></p><p><strong>(iii) Two particular students NOT together:</strong></p><p>Total ways - (ways with both together) = C(12,4) - C(10,2) = 495 - 45 = <strong>450</strong></p><p><strong>(iv) Two particular students together only:</strong></p><p>Case 1: Both included—fix them, select 2 from remaining 10: C(10,2) = 45</p><p>Case 2: Both excluded—select 4 from remaining 10: C(10,4) = 210</p><p>Total = 45 + 210 = <strong>255</strong></p><p><strong>(v) Students A,B refuse together AND Students C,D wish together only:</strong></p><p>Total ways - (A,B together) - (not both C,D) + (A,B together AND not both C,D)</p><p>= C(12,4) - [C(10,2)] - [C(12,4) - C(10,2)] + [C(8,2)]</p><p>= 495 - 45 - 450 + 36 = <strong>216</strong></p><p>OR: Valid cases = (Neither A nor B) + (Exactly one of A,B) with both C,D = C(10,4) + 2×C(1,1)×C(10,2) = 210 + 2×45 = 300, then subtract cases where C,D separated = 300 - 84 = <strong>216</strong></p>
Correct Answer: (i) 495, (ii) 45, (iii) 450, (iv) 255, (v) 216