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.
- 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 card numbered 25 has two red digits,
- the card numbered 26 has one red digit,
- the card numbered 148 has no red digits.
- By considering the cards with one digit, two digits and three digits, or otherwise, determine how many cards have no red digits.
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
- the card numbered 2 is in set A ,
- the card numbered 15 is in sets B and C ,
- the card numbered 23 is in none of the sets.
\includegraphics[max width=\textwidth, alt={}, center]{133395d2-5020-4054-a229-70168f1d0f95-4_588_1150_1667_246} - By considering the cards with one digit, two digits and three digits, or otherwise, determine how many cards in set A have no red digits.
- Given that, for the cards with no red digits, \(n ( B ) = 21 , n ( C ) = 9\) and \(n ( A \cap B ) = 12\), use the inclusion-exclusion principle to determine how many of the cards with no red digits are in none of the sets A, B or C.