Theory question 113
Permutations and Combinations(a) Simplify n+1C3 − n−1C3.
(b) A fair die is thrown five times. Calculate, correct to three decimal places, the probability of containing (i) at most two sixes; (ii) exactly three sixes.
Show the model answer
Model answer
(a) [(n + 1)n(n − 1) − (n − 1)(n − 2)(n − 3)]/6
= (n − 1)(6n − 6)/6 = (n − 1)2
(b) p = 1/6, q = 5/6, n = 5
(i) P(0) + P(1) + P(2) = 0.4019 + 0.4019 + 0.1608 = 0.965
(ii) 5C3(1/6)3(5/6)2 = 250/7776 = 0.032
Explanation
Write each combination as a product and take out the common factor (n − 1).