O-LevelAdditional MathematicsPermutations and combinations4037/12 · May/June 2023 · Q78 Marks

Find the number of ways in which 14 people can be put into 4 groups containing 2, 3, 4 and 5 people.

(a)Find the number of ways in which 14 people can be put into 4 groups containing 2, 3, 4 and 5 people.[3]
(b)(i)6-digit numbers are to be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each digit may be used only once in any 6-digit number. A 6-digit number must not start with 0. Find how many 6-digit numbers can be formed if there are no further restrictions[1]
(b)(ii)the 6-digit number is divisible by 10[1]
(b)(iii)the 6-digit number is greater than 500000 and even.[3]

Mark scheme

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

[B1] for a product of 3 combinations (ignore combinations that are equal to 1), one of which must be in the form 14Ck where k = 2, 3, 4, 5, 9, 10, 11, 12
[B2] for a correct product of combinations 14C2 x 12C3 x 9C4 oe, soi
[B1] 2 522 520
[B1] 136 080
[B1] 15 120
[B1] for 8P4 or 1680 or (8 × 7 × 6 × 5)
[B2] for 8 × 8P4 (13440) oe or 15 × 8P4 (25200) oe

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