Finding Set Cardinalities from Constraints

A question is this type if and only if it gives constraints about how many elements are in various set combinations and asks you to find the number in a specific region (often 'all three' or a particular intersection).

4 questions · Moderate -0.1

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OCR H240/02 Q11
8 marks Moderate -0.3
11 Each of the 30 students in a class plays at least one of squash, hockey and tennis.
  • 18 students play squash
  • 19 students play hockey
  • 17 students play tennis
  • 8 students play squash and hockey
  • 9 students play hockey and tennis
  • 11 students play squash and tennis
    1. Find the number of students who play all three sports.
A student is picked at random from the class.
  • Given that this student plays squash, find the probability that this student does not play hockey. Two different students are picked at random from the class, one after the other, without replacement.
  • Given that the first student plays squash, find the probability that the second student plays hockey.
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    OCR Further Discrete AS 2020 November Q2
    8 marks Moderate -0.8
    2 Jameela needs to store ten packages in boxes. She has a list showing the size of each package. The boxes are all the same size and Jameela can use up to six of these boxes to store all the packages.
    1. Which of the following is a question that Jameela could ask which leads to a construction problem? Justify your choice.
      The total volume of the packages is \(1 \mathrm {~m} ^ { 3 }\). The volume of each of the six boxes is \(0.25 \mathrm {~m} ^ { 3 }\).
    2. Explain why a solution to the problem of storing all the packages in six boxes may not exist. The volume of each package is given below.
      PackageABCDEFGHIJ
      Volume \(\left( \mathrm { m } ^ { 3 } \right)\)0.200.050.150.250.040.030.020.020.120.12
    3. By considering the five largest packages (A, C, D, I and J) first, explain what happens if Jameela tries to pack the 10 packages using only four boxes. You may now assume that the packages will always fit in the boxes if there is enough volume.
    4. Use first-fit to find a way of storing the packages in the boxes. Show the letters of the packages in each box, in the order that they are packed into that box. The order of the packages within a box does not matter and the order of the boxes does not matter. So, for example, having A and E in box 1 is the same as having E and A in box 2 , but different from having A in one box and E in a different box.
    5. Suppose that packages A and B are not in the same box. In this case the following are true:
      Use the inclusion-exclusion principle to determine how many of the 8 ways include neither package F nor package G.
    OCR Further Discrete 2018 March Q3
    8 marks Standard +0.8
    50 people are at a TV game show. 21 of the 50 are there to take part in the game show and the others are friends who are in the audience, 22 are women and 20 are from London, 2 are women from London who are there to take part in the game show and 15 are men who are not from London and are friends who are in the audience.
    1. Deduce how many of the 50 people are in two of the categories 'there to take part in the game show', 'is a woman' and 'is from London', but are not in all three categories. [3]
    The 21 people who are there to take part in the game show are moved to the stage where they are seated in two rows of seats with 20 seats in each row. Some of the seats are empty.
    1. Show how the pigeonhole principle can be used to show that there must be at least one pair of these 21 people with no empty chair between them. [2]
    The 21 people are split into three sets of 7. In each round of the game show, three of the people are chosen. The three people must all be from the same set of 7 but once two people have played in the same round they cannot play together in another round. For example, if A plays with B and C in round 1 then A cannot play with B or with C in any other round.
    1. By first considering how many different rounds can be formed using the first set of seven people, deduce how many rounds there can be altogether. [3]
    OCR H240/02 2017 Specimen Q11
    8 marks Moderate -0.3
    Each of the 30 students in a class plays at least one of squash, hockey and tennis. • 18 students play squash • 19 students play hockey • 17 students play tennis • 8 students play squash and hockey • 9 students play hockey and tennis • 11 students play squash and tennis
    1. Find the number of students who play all three sports. [3]
    A student is picked at random from the class.
    1. Given that this student plays squash, find the probability that this student does not play hockey. [1]
    Two different students are picked at random from the class, one after the other, without replacement.
    1. Given that the first student plays squash, find the probability that the second student plays hockey. [4]