Theory question 102
Permutations and CombinationsGiven that xC2 + xC3 − 4x = 0, where x is a positive integer, find the value of x.
Show the model answer
Model answer
x(x − 1)/2 + x(x − 1)(x − 2)/6 = 4x.
Divide by x (x ≠ 0) and multiply by 6: 3(x − 1) + (x − 1)(x − 2) = 24.
(x − 1)(x + 1) = 24, so x² = 25.
x = 5 (positive integer).
Explanation
Write ⁿCᵣ as factorials and cancel.