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Date November 2017 Marks available 3 Reference code 17N.2.hl.TZ0.9
Level HL only Paper 2 Time zone TZ0
Command term Find Question number 9 Adapted from N/A

Question

Twelve students are to take an exam in advanced combinatorics.
The exam room is set out in three rows of four desks, with the invigilator at the front of the room, as shown in the following diagram.

INVIGILATOR

Desk 1Desk 2Desk 3Desk 4Desk 5Desk 6Desk 7Desk 8Desk 9Desk 10Desk 11Desk 12

Two of the students, Helen and Nicky, are suspected of cheating in a previous exam.

Find the number of ways the twelve students may be arranged in the exam hall.

[1]
a.

Find the number of ways the students may be arranged if Helen and Nicky must sit so that one is directly behind the other (with no desk in between). For example Desk 5 and Desk 9.

[2]
b.

Find the number of ways the students may be arranged if Helen and Nicky must not sit next to each other in the same row.

[3]
c.

Markscheme

12! (=479001600)     A1

[1 mark]

a.

METHOD 1

8×2=16 ways of sitting Helen and Nicky, 10! ways of sitting everyone else     (A1)

16×10!

=58060800     A1

METHOD 2

8×1×10! (=29030400) ways if Helen sits in the front or back row

4×2×10! (=29030400) ways if Helen sits in the middle row     (A1)

 

Note:     Award A1 for one correct value.

 

2×29030400

=58060800     A1

[2 marks]

b.

METHOD 1

9×2×0! (=65318400) ways if Helen and Nicky sit next to each other     (A1)

attempt to subtract from total number of ways     (M1)

12!9×2×10!

=413683200     A1

METHOD 2

6×10×10! (=217728000) ways if Helen sits in column 1 or 4     (A1)

6×9×10! (=195955200) ways if Helen sits in column 2 or 3     (A1)

217728000+195955200

=413683200     A1

[3 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 1 - Core: Algebra » 1.3 » Counting principles, including permutations and combinations.

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