Exactly N letters between items

Questions requiring exactly a specific number of letters between two particular items, solved by fixing positions with the required gap.

4 questions

CAIE S1 2021 June Q6
2 marks
6
  1. How many different arrangements are there of the 11 letters in the word REQUIREMENT? [2]
  2. How many different arrangements are there of the 11 letters in the word REQUIREMENT in which the two Rs are together and the three Es are together?
  3. How many different arrangements are there of the 11 letters in the word REQUIREMENT in which there are exactly three letters between the two Rs?
    Five of the 11 letters in the word REQUIREMENT are selected.
  4. How many possible selections contain at least two Es and at least one R?
CAIE S1 2024 November Q6
6
  1. Find the number of different arrangements of the 9 letters in the word HAPPINESS.
  2. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the first and last letters are not the same as each other.
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  3. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the two Ps are together and there are exactly two letters between the two Ss.
    The 9 letters in the word HAPPINESS are divided at random into a group of 5 and a group of 4 .
  4. Find the probability that both Ps are in one group and both Ss are in the other group.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S1 2016 June Q7
7
  1. Find the number of different arrangements which can be made of all 10 letters of the word WALLFLOWER if
    1. there are no restrictions,
    2. there are exactly six letters between the two Ws.
  2. A team of 6 people is to be chosen from 5 swimmers, 7 athletes and 4 cyclists. There must be at least 1 from each activity and there must be more athletes than cyclists. Find the number of different ways in which the team can be chosen.
CAIE S1 2021 November Q4
4
  1. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
  2. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?