O-LevelAdditional MathematicsPermutations and combinations4037/12 · May/June 2015 · Q510 Marks

(a) A security code is to be chosen using 6 of the following: • the letters A, B and C • the numbers 2, 3 and 5 • the symbols * and $. None of the above may be used more than once. Find the number of different security codes that may be chosen if

(a)(i)there are no restrictions,[1]
(a)(ii)the security code starts with a letter and finishes with a symbol,[2]
(a)(iii)the two symbols are next to each other in the security code.[3]
(b)(i)Two teams, each of 4 students, are to be selected from a class of 8 boys and 6 girls. Find the number of different ways the two teams may be selected if there are no restrictions,[2]
(b)(ii)one team is to contain boys only and the other team is to contain girls only.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

[1] (a)(i) B1 20160
[2] (a)(ii) B1,B1 B1 for ⁶P₄ (must be seen in a product) B1 for all correct, with no further working
[3] (a)(iii) B1,B1 B1 for ⁵P₄ (must be seen in a product) B1 for 5 (must be in a product) B1 for all correct, with no further working
[3] (a)(iii) Alternative 1: B2 for ⁶C₄ × 5! B1 for ⁶C₄ × 5! ×2
[2] (b)(i) B1,B1 B1 for either ¹⁴C₄ or ¹⁴C₁₀ as part of a product B1 for correct answer, with no further working
[2] (b)(ii) B1,B1 B1 for either ⁸C₄ or ⁶C₄ as part of a product B1 for correct answer with no further working

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