A-Level Mathematics May/June 2024 Q7(c): Find the probability that a randomly chosen arrangement of the 10 letters in the word R…

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  The word is REGENERATE.

  There are 10 letters in total. Number of vowels = 5 (A, E, E, E, E). The letter 'E' is repeated 4 times. Number of consonants = 5 (R, G, N, R, T). The letter 'R' is repeated 2 times.

  First, we find the total number of distinct arrangements of the 10 letters. This was found in part (a). Total arrangements = $\frac{10!}{4! \times 2!} = \frac{3,628,800}{24 \times 2} = 75,600$.

  For the letters to alternate, the arrangement must be of the form CVCVCVCVCV or VCVCVCVCVC, where C is a consonant and V is a vowel. Since there are 5 consonants and 5 vowels, both patterns are possible.

  Step 1: Arrange the consonants. The 5 consonants are G, N, R, R, T. The number of ways to arrange the consonants is: $\frac{5!}{2!} = \frac{120}{2} = 60$.

  Step 2: Arrange the vowels. The 5 vowels are A, E, E, E, E. The number of ways to arrange the vowels is: $\frac{5!}{4!} = \frac{120}{24} = 5$.

  Step 3: Combine the arrangements. For any arrangement of consonants and any arrangement of vowels, they can be interleaved in two ways (starting with a consonant or starting with a vowel). Number of favourable arrangements = (Arrangements of consonants) $\times$ (Arrangements of vowels) $\times$ 2 $= (\frac{5!}{2!}) \times (\frac{5!}{4!}) \times 2$ $= 60 \times 5 \times 2 = 600$.

  Step 4: Calculate the probability. Probability = $\frac{\text{Number of favourable arrangements}}{\text{Total number of arrangements}}$

  Probability = $\frac{600}{75600}$

  Simplifying the fraction: Probability = $\frac{6}{756} = \frac{1}{126}$.

  The final probability is $\frac{1}{126}$.

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  How the marks are awarded

  • B1 — Awarded for correctly calculating the number of arrangements of the 5 consonants (G, N, R, R, T), accounting for the repeated 'R'. This is shown by the calculation 5!/2!.
  • B1 — Awarded for correctly calculating the number of arrangements of the 5 vowels (A, E, E, E, E), accounting for the four repeated 'E's. This is shown by the calculation 5!/4!.
  • M1 — Awarded for the method to find the total number of favourable (alternating) arrangements. This involves multiplying the consonant arrangements, the vowel arrangements, and the 2 possible starting patterns (CVCV... or VCVC...). This is shown by (5!/2!) × (5!/4!) × 2.
  • M1 — Awarded for the method to find the probability by dividing the number of favourable arrangements (the numerator) by the total number of possible arrangements (the denominator from 7(a)). This is shown by the fraction 600/75600.
  • A1 — Awarded for the correct final answer, correctly simplified to 1/126. An unsimplified but correct fraction like 600/75600 is also acceptable.

  Common mistakes

  • Forgetting to multiply by 2. Students calculate the arrangements for one pattern (e.g., CVCVCVCVCV) but forget that the arrangement can also start with a vowel, leading to an answer of 1/252.
  • Incorrectly handling repetitions in the permutations, for example by calculating 5! for consonants or 5! for vowels, or mixing up the denominators (e.g., 5!/4! for consonants and 5!/2! for vowels).
  • Using an incorrect total number of arrangements in the denominator, often due to an error in part (a) such as writing 10!/(4!+2!) or forgetting one of the repetitions.
  • Adding the number of consonant and vowel arrangements instead of multiplying them, showing a misunderstanding of the fundamental counting principle.

  Examiner tip: For arrangement problems with restrictions like 'alternating', calculate the permutations for each subgroup first, then consider the number of ways to position these subgroups relative to each other.

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