Kenneth ChengMathematical modeling
← Senior Mathematics Studio

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Arrange the same nine distinct books uniformly at random. Find the probability that books of each category occur together in one block.

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TOPIC 01

2021 JM01

The books remain distinct even within the same category.

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Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

Arrange the same nine distinct books uniformly at random. Find the probability that books of each category occur together in one block.

Official paper · jm01-2021 · II.1(b) · PDF 4

Official original and suggested answers ↗ · Suggested answer PDF page 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Treat each category as a block.
Hint 2
Count both block order and order inside each block.
Worked solution

  1. There are 3! block orders, and internal orders 4!,2!,3!.

    F=3! 4! 2! 3!F=3!\,4!\,2!\,3!

  2. Divide by all permutations of the distinct books.

    P=3!4!2!3!9!=1/210P=\frac{3!4!2!3!}{9!}=1/210

1/210.

Checks and common pitfalls: The books remain distinct even within the same category.

Reasoning checklist · self / teacher assessment
  • Teaching assessment checklist, independently authored. Use the original paper for official marks.
  • There are 3! block orders, and internal orders 4!,2!,3!.
    F=3! 4! 2! 3!F=3!\,4!\,2!\,3!
  • Divide by all permutations of the distinct books.
    P=3!4!2!3!9!=1/210P=\frac{3!4!2!3!}{9!}=1/210

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Curriculum and source notes ↗