Arrangements with adjacency requirements

A question is this type if and only if it requires certain items to be together (adjacent) in the arrangement.

16 questions

CAIE S1 2020 November Q7
7
  1. Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that all 3 Es are together.
  2. Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that the Ps are not next to each other.
  3. Find the probability that a randomly chosen arrangement of the 10 letters of the word SHOPKEEPER has an E at the beginning and an E at the end.
    Four letters are selected from the 10 letters of the word SHOPKEEPER.
  4. Find the number of different selections if the four letters include exactly one P .
    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 2022 November Q7
7
  1. Find the number of different arrangements of the 9 letters in the word ALLIGATOR in which the two As are together and the two Ls are together.
  2. The 9 letters in the word ALLIGATOR are arranged in a random order. Find the probability that the two Ls are together and there are exactly 6 letters between the two As.
  3. Find the number of different selections of 5 letters from the 9 letters in the word ALLIGATOR which contain at least one A 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 2002 June Q5
5 The digits of the number 1223678 can be rearranged to give many different 7 -digit numbers. Find how many different 7-digit numbers can be made if
  1. there are no restrictions on the order of the digits,
  2. the digits 1,3,7 (in any order) are next to each other,
  3. these 7 -digit numbers are even.
  4. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma , \mathrm { P } ( X > 3.6 ) = 0.5\) and \(\mathrm { P } ( X > 2.8 ) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\).
  5. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8.
CAIE S1 2020 June Q2
2
  1. Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the three Es are together and the two Ls are together.
  2. Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the two Ls are not next to each other.
CAIE S1 2007 June Q5
5
  1. Find the number of ways in which all twelve letters of the word REFRIGERATOR can be arranged
    (a) if there are no restrictions,
    (b) if the Rs must all be together.
  2. How many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no Rs and two Es?
CAIE S1 2015 November Q5
5
  1. Find the number of ways in which all nine letters of the word TENNESSEE can be arranged
    1. if all the letters E are together,
    2. if the T is at one end and there is an S at the other end.
  2. Four letters are selected from the nine letters of the word VENEZUELA. Find the number of possible selections which contain exactly one E .
CAIE S1 2019 November Q6
6
  1. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that all four Es are together.
  2. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that the Ss are not next to each other.
    Four letters are selected from the 12 letters of the word STEEPLECHASE.
  3. Find the number of different selections if the four letters include exactly one \(S\).
CAIE S1 2019 November Q7
7
  1. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that all three Os are together and both Ts are together.
  2. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that the Ts are not together.
  3. Find the probability that a randomly chosen arrangement of the 9 letters of the word TOADSTOOL has a T at the beginning and a T at the end.
  4. Five letters are selected from the 9 letters of the word TOADSTOOL. Find the number of different selections if the five letters include at least 2 Os and at least 1 T .
    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 Specimen Q5
5
  1. Find the number of ways in which all nine letters of the word TENNESSEE can be arranged
    1. if all the letters E are together,
    2. if the T is at one end and there is an S at the other end.
  2. Four letters are selected from the nine letters of the word VENEZUELA. Find the number of possible selections which contain exactly one E .
CAIE S1 2011 November Q6
6
  1. Find the number of different ways in which the 12 letters of the word STRAWBERRIES can be arranged
    1. if there are no restrictions,
    2. if the 4 vowels \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { I }\) must all be together.
    1. 4 astronauts are chosen from a certain number of candidates. If order of choosing is not taken into account, the number of ways the astronauts can be chosen is 3876 . How many ways are there if order of choosing is taken into account?
    2. 4 astronauts are chosen to go on a mission. Each of these astronauts can take 3 personal possessions with him. How many different ways can these 12 possessions be arranged in a row if each astronaut's possessions are kept together?
OCR S1 2008 January Q1
1
  1. The letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E are arranged in a straight line.
    (a) How many different arrangements are possible?
    (b) In how many of these arrangements are the letters A and B next to each other?
  2. From the letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , two different letters are selected at random. Find the probability that these two letters are A and B .
OCR S1 2006 June Q3
3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.
  1. How many different arrangements of the letters are possible?
  2. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
  3. Find the probability that at least one of these 2 cards has D printed on it.
OCR S1 Specimen Q3
3 Five friends, Ali, Bev, Carla, Don and Ed, stand in a line for a photograph.
  1. How many different possible arrangements are there if Ali, Bev and Carla stand next to each other?
  2. How many different possible arrangements are there if none of Ali, Bev and Carla stand next to each other?
  3. If all possible arrangements are equally likely, find the probability that two of Ali, Bev and Carla are next to each other, but the third is not next to either of the other two.
OCR S1 2010 January Q8
8 The five letters of the word NEVER are arranged in random order in a straight line.
  1. How many different orders of the letters are possible?
  2. In how many of the possible orders are the two Es next to each other?
  3. Find the probability that the first two letters in the order include exactly one letter E.
    \(9 R\) and \(S\) are independent random variables each having the distribution \(\operatorname { Geo } ( p )\).
  4. Find \(\mathrm { P } ( R = 1\) and \(S = 1 )\) in terms of \(p\).
  5. Show that \(\mathrm { P } ( R = 3\) and \(S = 3 ) = p ^ { 2 } q ^ { 4 }\), where \(q = 1 - p\).
  6. Use the formula for the sum to infinity of a geometric series to show that $$\mathrm { P } ( R = S ) = \frac { p } { 2 - p }$$
OCR S1 2011 June Q6
6 A group of 7 students sit in random order on a bench.
  1. (a) Find the number of orders in which they can sit.
    (b) The 7 students include Tom and Jerry. Find the probability that Tom and Jerry sit next to each other.
  2. The students consist of 3 girls and 4 boys. Find the probability that
    (a) no two boys sit next to each other,
    (b) all three girls sit next to each other.
SPS SPS FM Statistics 2026 January Q8
8. Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.
  1. How many different arrangements of the letters are possible?
  2. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
  3. Find the probability that at least one of these 2 cards has D printed on it.
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