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).

2 questions

OCR H240/02 Q11
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.
    •
    OCR Further Discrete AS 2020 November Q2
    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.
      • In how many different ways can I fit the packages in the boxes?
      • How can I fit the packages in the boxes?
      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:
      • there are 8 different ways to put any or none of \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H with A
      • 3 include F with A
      • 3 include G with A
      • 1 includes both F and G with A
      Use the inclusion-exclusion principle to determine how many of the 8 ways include neither package F nor package G.