A-LevelMathematicsProbability and statistics9709/51 · May/June 2024 · Q710 Marks

The eight digits 1, 2, 2, 3, 4, 4, 4, 5 are arranged in a line.

(a)How many different arrangements are there of these 8 digits?[1]
(b)Find the number of different arrangements of the 8 digits in which there is a 2 at the beginning, a 2 at the end and the three 4s are not all together.[4]
(c)Three digits are selected at random from the eight digits 1, 2, 2, 3, 4, 4, 4, 5. Find the probability that the three digits are all different.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B1] 8! / (2!3!) = 3360
[M1] 6! / 3! – x r – s, r = 1, 2 and s a positive integer (including 0).
[B1] 4! Seen either alone or in t – 4!, t an integer value > 24.
[M1] 6! / 3! – x r – 4! x u, r =1, 2 and u = 1, 2.
[A1] 96
[M1] One correct calculation, unsimplified for an identified scenario containing 1, 2 and/or 1, 4.
[A1] Two correct outcomes evaluated, accept unsimplified.
[M1] Four correct scenarios added.
[B1] Used as denominator of probability expression. C(8,3) [= 56]
[A1] 34/56 = 17/28 = 0.607

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.

Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, May/June 2024 Paper 5 Variant 1, question 7.

Questions and mark schemes are from past examination papers and remain the copyright of the exam board. How Oracle Prep builds its content