## Question 7(a):

$\{ABCDE\}$

$\{ABCE \mid D\}$ $\{ABDE \mid C\}$ $\{ACDE \mid B\}$
$\{ABE \mid CD\}$ $\{ACE \mid BD\}$ $\{ADE \mid BC\}$ | M1 | At least three correct partitions, in any unambiguous form

$\{ABE \mid C \mid D\}$ $\{ACE \mid B \mid D\}$ $\{ADE \mid B \mid C\}$
$\{AE \mid BCD\}$
$\{AE \mid BC \mid D\}$ $\{AE \mid BD \mid C\}$ $\{AE \mid B \mid CD\}$ | A1 | At least three correct partitions with different structures

$\{AE \mid B \mid C \mid D\}$ | A1 | All 15 correct with no extras or duplicates

**[3 marks total]**

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## Question 7(b):

Partition the E in $\{ABCDE\}$ to get $\{ABCD \mid E\}$

For each of the 7 partitions from part (a) into 2 subsets, put E in the other subset e.g. $\{ABCE \mid D\}$ becomes $\{ABC \mid DE\}$ | M1 | 1 partition with 1 subset and move E, or write down $\{ABCD \mid E\}$
 | A1 | 7 partitions with 2 subsets from part (a). Or listing these 7 partitions: $\{ABC \mid DE\}$ $\{ABD \mid CE\}$ $\{ACD \mid BE\}$ $\{AB \mid CDE\}$ $\{AC \mid BDE\}$ $\{AD \mid BCE\}$ $\{A \mid BCDE\}$

**Alternative method 2:**
$^4C_4 = 1 = \{ABCD\}$ hence $\{ABCD \mid E\}$
$^4C_3 = 4$ e.g. $\{ABC \mid D\}$ hence $\{ABC \mid DE\}$
e.g. $\{A \mid BCD\}$ hence $\{A \mid BCDE\}$
$^4C_2 \div 2 = 3$ e.g. $\{AB \mid CD\}$ hence $\{AB \mid CDE\}$ | M1 | Showing how E can be added to $\{ABCD\}$ to give $\{ABCD \mid E\}$
 | A1 | Explaining why partitions of $\{ABCD\}$ into 2 subsets generate 7 subsets of the required form

**Alternative method 3:**
A is in one subset and E in the other. For each of B, C and D there are 2 choices of subset: $2^3 = 8$ | M1 | Identifying where 2 and 3 come from
 | A1 | $2^3$ or equivalent explanation of given result

**Alternative method 4:**
Subsets of sizes 1 and 4 $\Rightarrow$ either A or E is on its own (and everything else together) $= 2$ ways. Subsets of sizes 2 and 3 $\Rightarrow$ smaller subset is one of A, E with one of B, C, D $= 2\times3 = 6$ ways | M1 | Explaining, without necessarily writing out the partitions, why there are 2 of this type (or 6 of the other type)
 | A1 | Explaining both types. Or variations on this using 1, 3, 3, 1 of the different types

**[2 marks total]**

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## Question 7(c)(i):

$\{ABCDE\}$ gives $\{ABCD \mid E\}$ only | B1 | Identify $\{ABCDE\}$ and show that it gives one new partition only

**[1 mark total]**

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## Question 7(c)(ii):

e.g. $\{ABE \mid C \mid D\}$ gives $\{AB \mid C \mid D \mid E\}$, $\{AB \mid CE \mid D\}$ and $\{AB \mid C \mid DE\}$ | B1 | Showing that a partition into more than 2 subsets will give more than 2 new partitions

**[1 mark total]**