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.

3 questions · Standard +0.8

5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems
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CAIE S1 2022 November Q6
10 marks Standard +0.8
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
4 marks Challenging +1.2
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. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-04_2718_38_107_2009}
SPS SPS FM Statistics 2021 June Q3
4 marks Standard +0.3
The letters of the word CHAFFINCH are written on cards.
  1. In how many ways can the letters be rearranged with no restrictions. [1]
  2. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them. [3]