Question

Among the $900$ residents of Aimeville, there are $195$ who own a diamond ring, $367$ who own a set of golf clubs, and $562$ who own a garden spade. In addition, each of the $900$ residents owns a bag of candy hearts. There are $437$ residents who own exactly two of these things, and $234$ residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.

Answer

Answer: 73

Concepts:inclusion-exclusion Venn Diagram double counting

Difficulty rating: 2010

Solution:

Adding the four ownership counts gives $195 + 367 + 562 + 900 = 2024$ item ownerships among the $900$ residents. Since everyone owns a bag of candy hearts, every resident owns at least one item, and a resident owning exactly $k$ items is counted $k - 1$ times beyond the first.

If $n_4$ residents own all four things, the extra counts total $\begin{aligned} 2024 - 900 &= 437 \cdot 1 + 234 \cdot 2 \\ &\quad {}+ n_4 \cdot 3, \end{aligned}$ so $1124 = 905 + 3 n_4,$ giving $n_4 = \frac{219}{3} = 73.$

2024 AIME II Problem 1

Answer: 73

Difficulty rating: 2010

Solution:

Problem 1 in Other Years