A group of 6 teenagers go boating. There are three boats available. One boat has room for 3 people, one has room for 2 people and one has room for 1 person. Find the number of different ways the group of 6 teenagers can be divided between the three boats.

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

1. [M1] ⁶C₃ × ³C₂ × ¹C₁ seen oe ⁶⁻ᵃ⁻ᵇC₆⁻ᵃ⁻ᵇ can be implied by 1 or omission, condone use of permutations, (for part (a))
2. [A1] Any correct method seen no addition/additional scenarios (for part (a))
3. [A1] Correct answer (for part (a))
4. [M1] A single expression with either 4!/3! × k or 3!/2! × k, k a positive integer seen oe (condone 2 identical expressions being added) (for part (b)(i))
5. [M1] Correctly multiplying their single expression by 2 or 2 identical expressions being added. (for part (b)(i))
6. [A1] Correct answer (for part (b)(i))
7. [B1] Total no of arrangements = 7!/(2!3!) = 420 (A) Accept unsimplified (for part (b)(ii))
8. [B1] No with 2s together = 6!/3! = 120 (B) Accept unsimplified (for part (b)(ii))
9. [M1] Subtraction indicated, possibly by their answer, no additional terms present (for part (b)(ii))
10. [A1] Exact value www (for part (b)(ii))