Arrangements with positional constraints

A question is this type if and only if it requires arranging letters/objects with specific items in fixed positions (e.g., 'E at beginning and end', 'R in central position').

14 questions

CAIE S1 2020 June Q6
6
  1. Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that there is an E at the beginning and an E at the end.
  2. Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that the Es are not together.
  3. Four letters are selected from the 10 letters of the word SUMMERTIME. Find the number of different selections if the four letters include at least one M and exactly one E .
CAIE S1 2020 June Q7
7
  1. Find the number of different possible arrangements of the 9 letters in the word CELESTIAL.
  2. Find the number of different arrangements of the 9 letters in the word CELESTIAL in which the first letter is C, the fifth letter is T and the last letter is E.
  3. Find the probability that a randomly chosen arrangement of the 9 letters in the word CELESTIAL does not have the two Es together.
    5 letters are selected at random from the 9 letters in the word CELESTIAL.
  4. Find the number of different selections if the 5 letters include at least one E and at most one L .
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2021 June Q6
6
  1. Find the total number of different arrangements of the 8 letters in the word TOMORROW.
  2. Find the total number of different arrangements of the 8 letters in the word TOMORROW that have an R at the beginning and an R at the end, and in which the three Os are not all together.
    Four letters are selected at random from the 8 letters of the word TOMORROW.
  3. Find the probability that the selection contains at least one O and at least one R .
CAIE S1 2024 June Q7
7 The eight digits \(1,2,2,3,4,4,4,5\) are arranged in a line.
  1. How many different arrangements are there of these 8 digits?
  2. Find the number of different arrangements of the 8 digits in which there is a 2 at the beginning, a 2 at the end and the three 4 s are not all together.
    Three digits are selected at random from the eight digits \(1,2,2,3,4,4,4,5\).
  3. Find the probability that the three digits are all different.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2024 June Q6
6
  1. How many different arrangements are there of the 9 letters in the word RECORDERS?
  2. How many different arrangements are there of the 9 letters in the word RECORDERS in which there is an E at the beginning, an E at the end and the three Rs are not all together?
    \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-12_2725_40_136_2007}
    The 9 letters of the word RECORDERS are divided at random into two groups: a group of 5 letters and a group of 4 letters.
  3. Find the probability that the three Rs are in the same group.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-14_2715_35_143_2012}
CAIE S1 2021 March Q6
4 marks
6
  1. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR.
  2. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR in which there is an R at the beginning and an R at the end, and the two As are not together. [4]
  3. Find the total number of different selections of 6 letters from the 11 letters of the word CATERPILLAR that contain both Rs and at least one A and at least one L.
CAIE S1 2024 November Q7
4 marks
7
  1. How many different arrangements are there of the 9 letters in the word INTELLECT in which the two Ts are together?
  2. How many different arrangements are there of the 9 letters in the word INTELLECT in which there is a T at each end and the two Es are not next to each other?
    Four letters are selected at random from the 9 letters in the word INTELLECT.
    [0pt]
  3. Find the percentage of the possible selections which contain at least one E and exactly one T. [4]
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-14_2715_31_106_2016}
CAIE S1 2015 June Q6
6
  1. Find the number of different ways the 7 letters of the word BANANAS can be arranged
    1. if the first letter is N and the last letter is B ,
    2. if all the letters A are next to each other.
  2. Find the number of ways of selecting a group of 9 people from 14 if two particular people cannot both be in the group together.
CAIE S1 2016 June Q6
6 Find the number of ways all 9 letters of the word EVERGREEN can be arranged if
  1. there are no restrictions,
  2. the first letter is R and the last letter is G ,
  3. the Es are all together. Three letters from the 9 letters of the word EVERGREEN are selected.
  4. Find the number of selections which contain no Es and exactly 1 R .
  5. Find the number of selections which contain no Es.
CAIE S1 2019 March Q7
7 Find the number of different arrangements that can be made of all 9 letters in the word CAMERAMAN in each of the following cases.
  1. There are no restrictions.
  2. The As occupy the 1st, 5th and 9th positions.
  3. There is exactly one letter between the Ms.
    Three letters are selected from the 9 letters of the word CAMERAMAN.
  4. Find the number of different selections if the three letters include exactly one M and exactly one A.
  5. Find the number of different selections if the three letters include at least one M.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2007 November Q3
3 The six digits 4, 5, 6, 7, 7, 7 can be arranged to give many different 6-digit numbers.
  1. How many different 6-digit numbers can be made?
  2. How many of these 6-digit numbers start with an odd digit and end with an odd digit?
CAIE S1 2013 November Q6
6 The 11 letters of the word REMEMBRANCE are arranged in a line.
  1. Find the number of different arrangements if there are no restrictions.
  2. Find the number of different arrangements which start and finish with the letter M .
  3. Find the number of different arrangements which do not have all 4 vowels ( \(\mathrm { E } , \mathrm { E } , \mathrm { A } , \mathrm { E }\) ) next to each other. 4 letters from the letters of the word REMEMBRANCE are chosen.
  4. Find the number of different selections which contain no Ms and no Rs and at least 2 Es.
CAIE S1 2019 November Q2
2
  1. How many different arrangements are there of the 9 letters in the word CORRIDORS?
  2. How many different arrangements are there of the 9 letters in the word CORRIDORS in which the first letter is D and the last letter is R or O ?
OCR S1 2012 June Q7
7
  1. 5 of the 7 letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) are arranged in a random order in a straight line.
    (a) How many different arrangements of 5 letters are possible?
    (b) How many of these arrangements end with a vowel (A or E)?
  2. A group of 5 people is to be chosen from a list of 7 people.
    (a) How many different groups of 5 people can be chosen?
    (b) The list of 7 people includes Jill and Jo. A group of 5 people is chosen at random from the list. Given that either Jill and Jo are both chosen or neither of them is chosen, find the probability that both of them are chosen.