A-LevelMathematicsProbability and statistics9709/51 · Oct/Nov 2023 · Q610 Marks

In a restaurant, the tables are rectangular. Each table seats four people: two along each of the longer sides of the table (see diagram). Eight friends have booked two tables, X and Y. Rajid, Sue and Tan are three of these friends. [Figure on page 12]

(a)The eight friends will be divided into two groups of 4, one group for table X and one group for table Y. Find the number of ways in which this can be done if Rajid and Sue must sit at the same table as each other and Tan must sit at the other table.[3]
(b)When the friends arrive at the restaurant, Rajid and Sue now decide to sit at table X on the same side as each other. Tan decides that he does not mind at which table he sits. Find the number of different seating arrangements for the 8 friends.[3]
(c)As they leave the restaurant, the 8 friends stand in a line for a photograph. Find the number of different arrangements if Rajid and Sue stand next to each other, but neither is at an end of the line.[4]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[M1] ⁵C₂ × r, r = positive integer, 1 implied, no addition.
[M1] s × 2, s = ⁵C₂ or ⁵P₂ or if ⁵C₂ or ⁵P₂ not present, s = a single integer > 1 or t! × 2, 2 ≤ t ≤ 8, no other terms.
[A1] 20
[M1] ⁶C₂ × 2 × 2 × t, t = positive integer ≥ 1.
[M1] ⁶P₂ × 2 × 2 × t, t = positive integer ≥ 1.
[M1] u × 4!, u = positive integer > 1.
[A1] If A0 scored, SC B1 for 2880 nfww.
[M1] 6! × v, v = positive integer ≥ 1.
[M1] w × 2 × 2, w = positive integer > 1. condone w × 4, w = positive integer > 1.
[A1] If A0 scored, SC B1 for 2880 nfww.
[M1] 7!×2- a, a = positive integer > 1.
[M1] b-6!×4, b = positive integer > 2880.
[M1] 7!×c- 6! × d, c = 1,2 and d = 1, 4.
[A1] If A0 scored, SC B1 for 7200 nfww.
[M1] 6!x ex f, e, f = positive integers ≥ 1.
[M1] 6! × 2 × f, f= positive integer ≥ 1. If 5! Used, SC B1 5! × 2 × f, f = positive integer > 1.
[M1] 6! xex 5, e = positive integer ≥ 1.
[A1] If A0, scored SC B1 for 7200 nfww.
[M1] 6P2 x ex f, e, f = positive integers ≥ 1.
[M1] 6P2 × 5! × f, f = positive integer ≥ 1. Condone 6C2 × 5! × f, f= positive integer ≥ 1.
[M1] 6P2 x ex 2, e = positive integer ≥ 1. Condone 6C2 × e × 2, e = positive integer ≥ 1.
[A1] If A0 scored, SC B1 for 7200 nfww.
[M1] 6Pn × a ×(6-n)!, a = positive integer, 1 < n < 5 seen once.
[M1] 6Pn × 2 × (6-n)!, a = positive integer, 1 < n < 5 seen at least 3 times in identified scenarios.
[M1] Add 5 values of appropriate scenarios only. No additional, incorrect or repeated scenarios. Accept unsimplified.
[A1] If A0 scored, SC B1 for 7200 nfww.