**Text Transcription for Educational Website:**---**Problem:**Airlines would like to board passengers in the order of decreasing seat numbers (large seat number first, second largest next, and so on), but passengers don’t like this policy and refused to go along. If two passengers randomly board a plane but probability that they board in order of decreasing seat numbers is 1/2. If three passengers randomly board the plane, the probability that they board in order of decreasing seat number is 1/6; if four passengers randomly board a plane the probability they board in order of decreasing seat number is 1/24; and if five passengers randomly board a plane, the probability they board in order of decreasing seat number is 1/120. Use the sequence 1/2, 1/6, 1/24, 1/120, ... as your guide.a) Determine the probability that if six passengers randomly board a plane they board in order of decreasing seat numbers. Enter your answer as a fraction.[Input Box] [Check Button]b) Determine the probability that if 12 passengers randomly board a plane, they board in order of decreasing seat numbers. Enter your answer as a fraction.[Input Box] [Check Button]**Diagram Explanation:**The problem contains input boxes where students can fill in their answers and check if they are correct by clicking on the check button. The sequence indicates a factorial pattern in probabilities.---

The information provided in the question are as follows : Probability that 2 passengers board the plane = 1/2 Probability that 3 passengers board the plane = 1/6 Probability that 4 passengers board the plane = 1/24 And, probability that 5 passengers board the plane = 1/120On looking at the sequence provided i.e., 1/2, 1/6, 1/24, 1/120, it can be seen that the given probabilities are of the form, P(n) = 1/n! ...(1) Where, n = number of passengers boarding the plane. n! = n × (n–1) × (n–2) × (n–3).....× 1 EXPLANATION: Probability that 2 passengers board the plane : Substituting, n = 2 in (1) we get, P(2) = 1/2! P(2) = 1/(2×1) ⇒P(2) = 1/2 Which is equal to the given probability for 2 passengers. Probability that 3 passengers board the plane : Substituting, n = 3 in (1) we get, P(3) = 1/3! P(3) = 1/(3×2×1) ⇒P(3) = 1/6 Same as the given probability for 3 passengers. Probability that 4 passengers board the plane : Substituting, n = 4 in (1) we get, P(4) = 1/4! P(4) = 1/(4×3×2×1) ⇒P(4) = 1/24 Which is equal to the given probability for 4 passengers. Probability that 5 passengers board the plane : Substituting, n = 5 in (1) we get, P(5) = 1/2! P(5) = 1/(5×4×3×2×1) ⇒P(5) = 1/120 Which is equal to the given probability for 5 passengers. Part (a) : Now, the probability that 6 passengers board the plane is calculated as follows : Substituting, n = 6 in the formula given by (1) provided in step 2 we get, P(6) = 1/6! P(6) = 1/(6×5×4×3×2×1) P(6) = 1/720 (ANSWER) Part (B) : The probability that 12 passengers board the plane is calculated as follows : Now, on substituting n = 12 in the formula (1) provided in step 2 we have, P(12) = 1/12! P(12) = 1/(12×11×10×9×8×7×6×5×4×3×2×1) P(12) = 1 / 479001600 (ANSWER) Part (A) : 1 / 720 Part (B) : 1 / 479001600