WAEC Further Mathematics 2012 Theory question 107

Theory question 107

Permutations and Combinations

Simplify ⁿ⁺¹C₄ − ⁿ⁻¹C₄

Show the model answer

Model answer

ⁿ⁺¹C₄ = (n + 1)n(n − 1)(n − 2)/24, ⁿ⁻¹C₄ = (n − 1)(n − 2)(n − 3)(n − 4)/24
Difference = (n − 1)(n − 2)[n(n + 1) − (n − 3)(n − 4)]/24
= (n − 1)(n − 2)(8n − 12)/24
= (n − 1)(n − 2)(2n − 3)/6 = (2n³ − 9n² + 13n − 6)/6

Explanation

Take out the common factor (n − 1)(n − 2) before expanding.