Challenging +1.8 This is a challenging combinatorics/probability problem requiring students to set up and solve an inequality involving factorials and permutations. Students must recognize that acceptable outcomes are 1 correct arrangement plus n(n-1)/2 arrangements with one adjacent transposition, then solve (1 + n(n-1)/2)/n! < 0.01. While the setup requires insight into counting permutations, the algebraic manipulation is straightforward once the inequality is established. The 7-mark allocation and need for both combinatorial reasoning and inequality solving place it well above average difficulty.
Alex claims that he can read people's minds. A volunteer, Jane, arranges the integers 1 to \(n\) in an order of Jane's own choice and Alex tells Jane what order he believes was chosen.
They agree that Alex's claim will be accepted if he gets the order completely correct or if he gets the order correct apart from two numbers which are the wrong way round.
They use a value of \(n\) such that, if Alex chooses the order of the integers at random, the probability that Alex's claim will be accepted is less than 1%.
Determine the smallest possible value of \(n\). [7]
Alex claims that he can read people's minds. A volunteer, Jane, arranges the integers 1 to $n$ in an order of Jane's own choice and Alex tells Jane what order he believes was chosen.
They agree that Alex's claim will be accepted if he gets the order completely correct or if he gets the order correct apart from two numbers which are the wrong way round.
They use a value of $n$ such that, if Alex chooses the order of the integers at random, the probability that Alex's claim will be accepted is less than 1%.
Determine the smallest possible value of $n$. [7]
\hfill \mbox{\textit{SPS SPS ASFM 2020 Q12 [7]}}