At least/at most N letters between items

Questions requiring at least or at most a specific number of letters between items, typically solved by casework or complementary counting.

2 questions

CAIE S1 2022 November Q6
6
  1. Find the number of different arrangements of the 9 letters in the word ACTIVATED.
  2. Find the number of different arrangements of the 9 letters in the word ACTIVATED in which there are at least 5 letters between the two As.
    Five letters are selected at random from the 9 letters in the word ACTIVATED.
  3. Find the probability that the selection does not contain more Ts than As.
CAIE S1 2024 November Q2
2
  1. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC.
  2. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC in which there are no more than two letters between the two As.
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