JCCDQBHWHCB015 Combinatorics : Conditional Over-counting

Introduction

     This question is particularly challenging, because it does not conform to any of the usual types of problems taught at ‘A’ level.  However, we can still use the following problem solving heuristics:-

     · drawing a diagram

     · considering a simpler problem first

Suggested Approach and Solution:-

     Consider first the case for 3 couples and draw a diagram with arrows.

Suggested Approach and Solution:-

     We use the same problem-solving tactics as for question 11.  Consider first the case for 3 couples and draw a diagram with arrows.

 

We now discover that this problem is a little more complicated than that of question 11. 

Each of the 3 men has arrows coming out to 2 x 3 – 2 = 4 persons (everyone except himself and his wife).  So there are 3 x (2 x 3 – 2) = 12 arrows.  There is double-counting (arrows in both directions), but only among the men.  So we need to subtract the number of lines connecting i.e.

³C₂ = 3 pairs of men.  Thus for the case of 3 couples, the calculation is:

          3 x (2 x 3 – 2) – ³C₂ = 12 – 3 = 9

We verify by counting from the diagram that this is correct.

For 13 couples, we have

          13 x (2 x 13 – 2) – ¹³C₂ = 247 ways.

         

FYI, the general formula is

          n x (2n – 2) – ⁿC₂