A-LevelMathematicsProbability and statistics9709/51 · Oct/Nov 2022 · Q610 Marks

A Social Club has 15 members, of whom 8 are men and 7 are women. The committee of the club consists of 5 of its members. The 15 members are having their photograph taken. They stand in three rows, with 3 people in the front row, 5 people in the middle row and 7 people in the back row.

(a)Find the number of different ways in which the committee can be formed from the 15 members if it must include more men than women.[4]
(b)In how many different ways can the 15 members of the club be divided into a group of 3, a group of 5 and a group of 7?[3]
(c)In one photograph Abel, Betty, Cally, Doug, Eve, Freya and Gino are the 7 members in the back row. In how many different ways can these 7 members be arranged so that Abel and Betty are next to each other and Freya and Gino are not next to each other?[3]

Mark scheme

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

[M1] ⁸Cₓ × ⁷C₅₋ₓ for x = 1, 2, 3, 4,or 5
[B1] Outcome for 4M1W or 3M2W correct and identified, accept unsimplified.
[M1] Add 3 values of appropriate scenarios, no incorrect scenarios, no repeated scenarios, accept unsimplified. Addition may be implied by final answer.
[A1] Value stated WWW.
[M1] ¹⁵Cᵣ × q, r = 3, 5, 7; q a positive integer >1
[M1] ¹⁵C₃ × ¹²C₅ [× ⁷C₇] s = 3,5,7; t = 3,5,7 ≠ s; u = 3,5,7 ≠ s,t
[A1] Final answer. If A0 awarded SC B1 for final answer 360360.
[M1] a! × 2! × b, a = 5, 6; b = 1,2 seen.
[M1] Either 6! × 2 - c, 1 < c < 1440 or d - 5! × 2 × 2, 1440 < d
[A1] 960

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.