A-LevelMathematicsProbability and statistics9709/51 · Oct/Nov 2021 · Q510 Marks

Raman and Sanjay are members of a quiz team which has 9 members in total. Two photographs of the quiz team are to be taken. For the first photograph, the 9 members will stand in a line.

(a)How many different arrangements of the 9 members are possible in which Raman will be at the centre of the line?[1]
(b)How many different arrangements of the 9 members are possible in which Raman and Sanjay are not next to each other?[3]
(c)In how many different ways can the 9 members be divided into a group of 5 and a group of 4?[2]
(d)For a random division into a group of 5 and a group of 4, find the probability that Raman and Sanjay are in the same group as each other.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[B1] [8! =] 40 320 Evaluated, exact value only.
[M1] 7! × k seen, k an integer > 1.
[M1] m×n(n-1) or m×"C2 or m×"P2, n = 7, 8 or 9, m an integer > 1.
[A1] Exact value only. SC B1 for final answer 282 240 WWW.
[M1] 9!- k, k an integer < 362 880 .
[M1] m-8! × n, m an integer > 40 320, n = 1,2.
[A1] Exact value only. SC B1 for final answer 282 240 WWW.
[M1] 9C5 [× 4C4] Cx [x9xC9-x,] x = 4, 5. Condone × 1 for 9-*C9-x•. Condone use of P.
[A1] 126 WWW
[M1] [Number of ways with Raman and Sanjay together on back row =] 7C3. Cx seen, x = 3 or 2.
[M1] [Number of ways with Raman and Sanjay together on front row =] 7C2. Summing two correct scenarios.
[A1] 56 Evaluated – may be seen used in probability. If M0 scored, SC B1 for 56 WWW.
[B1 FT] Probability = their 56 / their (c) = 56 / 126 = 4/9 = 0.444. FT their 56 from adding 2 or more scenarios in numerator and their (c) or correct as denominator.

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