Pre-U Pre-U 9794/3 2016 June — Question 5 4 marks

Exam BoardPre-U
ModulePre-U 9794/3 (Pre-U Mathematics Paper 3)
Year2016
SessionJune
Marks4
TopicCombinations & Selection
TypeArrangements of letters with restrictions
DifficultyModerate -0.3 This is a straightforward probability question using permutations with repeated letters. Students need to count total arrangements (8!/3!2!), subtract arrangements where the two A's are adjacent (treating AA as one unit: 7!/3!), and find the probability. The technique is standard and commonly practiced, making it slightly easier than average, though the repeated letters add minor complexity.
Spec5.01a Permutations and combinations: evaluate probabilities
The letters of the word 'SEPARATE' are to be rearranged. Find the probability that, in a randomly chosen rearrangement, the two letters 'A' are not next to each other. [4]
AnswerMarks Guidance
\(\frac{8!}{2! 2!} (= 10080)\)B1 Arrangements of SEPARATE.
ALT version 1:
AnswerMarks Guidance
\(\frac{6!}{2!} \times \mathrm{C}_2 (= 360 \times 21 = 7560)\)B1 Arrangements of SEPBTE and place As apart.
\(p = \frac{6!(2!) \times {^7}\mathrm{C}_2}{8!(2! 2!)}\)M1 Ratio of attempts at relevant expressions.
\(= \frac{3}{4}\)A1 c.a.o.
ALT version 2:
AnswerMarks Guidance
\(\frac{7!}{2!} (= 2520)\)B1 Arrangements with As together.
\(p = 1 - \frac{7!(2!)}{8!(2! 2!)}\)M1 \(1 -\) ratio of attempts at relevant expressions.
\(= \frac{3}{4}\)A1 c.a.o. 
$\frac{8!}{2! 2!} (= 10080)$ | B1 | Arrangements of SEPARATE.

**ALT version 1:**

$\frac{6!}{2!} \times \mathrm{C}_2 (= 360 \times 21 = 7560)$ | B1 | Arrangements of SEPBTE and place As apart.

$p = \frac{6!(2!) \times {^7}\mathrm{C}_2}{8!(2! 2!)}$ | M1 | Ratio of attempts at relevant expressions.

$= \frac{3}{4}$ | A1 | c.a.o. | [4]

**ALT version 2:**

$\frac{7!}{2!} (= 2520)$ | B1 | Arrangements with As together.

$p = 1 - \frac{7!(2!)}{8!(2! 2!)}$ | M1 | $1 -$ ratio of attempts at relevant expressions.

$= \frac{3}{4}$ | A1 | c.a.o.

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The letters of the word 'SEPARATE' are to be rearranged. Find the probability that, in a randomly chosen rearrangement, the two letters 'A' are not next to each other. [4]

\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q5 [4]}}
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