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O-LevelAdditional MathematicsPermutations and combinations4037/21 · May/June 2015 · Q58 Marks

There are twenty numbered balls in a bag. Two of the balls are numbered 0, six are numbered 1, five are numbered 2 and seven are numbered 3, as shown in the table below. Number on ball (0, 1, 2, 3), Frequency (2, 6, 5, 7). Four of these balls are chosen at random, without replacement. Calculate the number of ways this can be done so that

  1. (a)A lock can be opened using only the number 4351. State whether this is a permutation or a combination of digits, giving a reason for your answer.[1]
  2. (b)(i)the four balls all have the same number,[2]
  3. (b)(ii)the four balls all have different numbers,[2]
  4. (b)(iii)the four balls have numbers that total 3.[3]

Mark scheme

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

  1. [B1] Permutation because the order matters oe
  2. [M1] ⁶C₄ + ⁵C₄ + ⁷C₄
  3. [A1] 55
  4. [M1] ²C₁ x ⁶C₁ x ⁵C₁ x ⁷C₁
  5. [A1] 420
  6. [M1] for either correct product
  7. [M1] adding two correct products
  8. [A1] 70 If 0 scored, then SC1 for 1,1,1,0 and 0,0,2,1 seen

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About this question

Topic
Permutations and combinations, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source
O-Level Additional Mathematics, May/June 2015 Paper 2 Variant 1, question 5.

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