# Carrotsticks' 2016 HSC MX2 Solutions (1 Viewer)

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#### Glyde

##### Member
To be uploaded here. I will initially only do Q15 and 16, and then complete the remaining questions tonight when I return home from an event.

Am currently at Mittagong Maccas if anybody wants to drop by!
i work there haha

cool

#### wu345

##### Member
no need to do the last part of Q16 because it was in the 2014 BOS Trial

#### InteGrand

##### Well-Known Member
no need to do the last part of Q16 because it was in the 2014 BOS Trial
Also the Year 12 3U Pender Textbook had some extension Q's on derangements, but I can't remember whether they got you to derive the formula for D(n) too (I suspect they did, but they didn't guide you through it as much as this HSC paper).

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#### raghib483

##### New Member
the relevant Cambridge question about derangements: its from exercise 10E

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#### Carrotsticks

##### Retired
Uploaded solutions in the original post!

#### InteGrand

##### Well-Known Member
Uploaded solutions in the original post!
$\bg_white \noindent Thanks for taking the trouble to write these up!$

$\bg_white \noindent Really minor thing, but for 16 (c) (iv), you used an initial condition of D(0) = 1. But the question as written only ever defined D(n) for n\geq 1, giving D(1) = 0 and D(2) = 1 as the initial conditions. (Of course, when derangements are defined starting from 0, we do set D(0) = 1, but the Q. here hasn't defined a D(0), so I think they wanted n=1,2 as the initial conditions. We could obtain a value of D(0) using the recurrence though so that D(2) would be 1, and it'd be D(0) = 1. This is indeed why the convention is to set D(0) = 1 if we define a derangement at 0.)$

$\bg_white \noindent The proof without using D(0) ends up being essentially the same though. From T_{n+1} = -T_{n}, we get T_{n} = (-1)^{n-2}T_2 = (-1)^n T_2, for n\geq 2 (i.e. n>1). Using the given initial conditions, T_{2} = D(2) -2D(1) = 1-0 = 1, so we have T_n = (-1)^n again. The HSC might not have cared if D(0) was used (not sure), but I guess a lot of students wouldn't have considered D(0) in the first place.$

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#### calamebe

##### Active Member
Also for 14) a) iii) you forgot the squared on the pi, again fairly minor thing.

#### Carrotsticks

##### Retired
Have fixed the errors you have mentioned. Thank you for pointing them out. The updated dropbox link is now in the original post.