7.01k Inclusion-exclusion: for two sets

1 Jane wants to travel from home to the local town. Jane can do this by train, by bus or by both train and bus.

1. Give an example of a problem that Jane could be answering that would give a construction problem. A website gives Jane all the possible buses and trains that she could use.
   Jane finds 7 possible ways to make the journey.

1 Four children, A, B, C and D, discuss how many of the 23 birthday parties held by their classmates they had gone to. Each party was attended by at least one of the four children. The results are shown in the Venn diagram below. \includegraphics[max width=\textwidth, alt={}, center]{50697293-6cdc-475f-981f-71a351b0ff5a-2_387_618_589_246}

1. Construct a complete graph \(\mathrm { K } _ { 4 }\), with vertices representing the four children and arcs weighted to show the number of parties each pair of children went to.
2. State a piece of information about the number of parties the children went to that is shown in the Venn diagram but is not shown in the graph. A fifth child, E, also went to some of the 23 parties shown in the Venn diagram.
   Every party that E went to was also attended by at least one of \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .

3 Six people play a game with 150 cards. Each player has a stack of cards in front of them and the remainder of the cards are in another stack on the table.

1. Use the pigeonhole principle to explain why at least one of the stacks must have at least 22 cards in it. The set of cards is numbered from 1 to 150 . Each digit '2', '3' and '5', whether as a units digit or a tens digit, is coloured red.
   So, for example
   The cards are put into a Venn diagram with three intersecting sets: \(\mathrm { A } = \{\) cards with a number that is a multiple of \(2 \}\) \(\mathrm { B } = \{\) cards with a number that is a multiple of \(3 \}\) \(\mathrm { C } = \{\) cards with a number that is a multiple of \(5 \}\) For example

1

1. (a) Show that the number of arrangements of 25 distinct objects is an integer multiple of \(5 ^ { 6 }\).
   (b) Explain how this shows that the number of arrangements of 25 distinct objects is a whole number of millions.
2. (a) Calculate the values of