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Date	May 2021	Marks available	4	Reference code	21M.1.AHL.TZ1.9
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	9	Adapted from	N/A

Question

A farmer has six sheep pens, arranged in a grid with three rows and two columns as shown in the following diagram.

Five sheep called Amber, Brownie, Curly, Daisy and Eden are to be placed in the pens. Each pen is large enough to hold all of the sheep. Amber and Brownie are known to fight.

Find the number of ways of placing the sheep in the pens in each of the following cases:

Each pen is large enough to contain five sheep. Amber and Brownie must not be placed in the same pen.

[4]

Each pen may only contain one sheep. Amber and Brownie must not be placed in pens which share a boundary.

[4]

Markscheme

METHOD 1

B has one less pen to select (M1)

EITHER

A and B can be placed in $6 \times 5$ ways (A1)

C, D, E have $6$ choices each (A1)

A (or B), C, D, E have $6$ choices each (A1)

B (or A) has only $5$ choices (A1)

THEN

$5 \times 6^{4} (= 6480)$ A1

METHOD 2

total number of ways $= 6^{5}$ (A1)

number of ways with Amber and Brownie together $= 6^{4}$ (A1)

attempt to subtract (may be seen in words) (M1)

$6^{5} - 6^{4}$

$= 5 \times 6^{4} (= 6480)$ A1

[4 marks]

METHOD 1

total number of ways $= 6! (= 720)$ (A1)

number of ways with Amber and Brownie sharing a boundary

$= 2 \times 7 \times 4! (= 336)$ (A1)

attempt to subtract (may be seen in words) (M1)

$720 - 336 = 384$ A1

METHOD 2

case 1: number of ways of placing A in corner pen

$3 \times 4 \times 3 \times 2 \times 1$

Four corners total no of ways is $4 \times (3 \times 4 \times 3 \times 2 \times 1) = 12 \times 4! (= 288)$ (A1)

case 2: number of ways of placing A in the middle pen

$2 \times 4 \times 3 \times 2 \times 1$

two middle pens so $2 \times (2 \times 4 \times 3 \times 2 \times 1) = 4 \times 4! (= 96)$ (A1)

attempt to add (may be seen in words) (M1)

total no of ways $= 288 + 96$

$= 16 \times 4! (= 384)$ A1

[4 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » AHL 1.10—Perms and combs, binomial with negative and fractional indices

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