A-LevelMathematicsProbability and statistics9709/53 · May/June 2022 · Q711 Marks

A group of 15 friends visit an adventure park. The group consists of four families. • Mr and Mrs Kenny and their four children • Mr and Mrs Lizo and their three children • Mrs Martin and her child • Mr and Mrs Nantes The group travel to the park in three cars, one containing 6 people, one containing 5 people and one containing 4 people. The cars are driven by Mr Lizo, Mrs Martin and Mr Nantes respectively.

(a)In how many different ways can the remaining 12 members of the group be divided between the three cars?[3]
(b)In how many different orders can the 15 friends go through the gate if Mr Lizo goes first and each family stays together?[3]
(c)In how many ways can the team be chosen from the group of 15 so that the 3 children are all from different families?[2]
(d)In how many ways can the team be chosen so that at least one of Mr Kenny or Mr Lizo is included?[3]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[M1] ¹²C_s x q, r = 3, 4, 5 q a positive integer > 1, no + or -.
[M1] ¹²C_s x ¹²⁻ˢC_t [x ¹²⁻ˢ⁻ᵗC_u] s = 3, 4, 5; t = 3, 4, 5 ≠ s; u = 3, 4, 5 ≠ s t
[A1] [792 × 35 =] 27 720. CAO
[M1] Product involving at least 3 of 4!, 6!, 2!, 2!.
[M1] w×3!, w integer > 1.
[A1] 414 720. WWW CAO
[M1] ⁷C₄ (adults) × ⁴C₁ × ³C₁.
[A1] 420
[M1] ⁵C₃ (or P₃) x c for one of the products or ⁶C₃ (or P₃) x c, positive integer >1 for first 2 products only.
[M1] Add 2 or 3 correct scenarios only values, no additional incorrect scenarios, no repeated scenarios. Accept unsimplified.
[A1] [Total or Difference=] 1680

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