Question

A survey of 500 television viewers produced the following information: 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, and 50 do not watch any of the three games. The number of viewers who watch \emph{exactly one of the three games is:}

• A. 325
• B. 310
• C. 315
• D. 372

Hint:

In set problems: \[ \text{Exactly one} = (\text{At least one}) - (\text{At least two}) \] Always subtract viewers who watch multiple categories when asked for \emph{exactly one}.

Answer 1

The Correct Option is C

Step 1: Find the number of viewers who watch at least one game. \[ \text{Total viewers} = 500 \] \[ \text{Viewers who watch none} = 50 \] \[ \Rightarrow \text{Viewers who watch at least one game} = 500 - 50 = 450 \]
Step 2: Count viewers who watch at least two games. Given: \[ \text{Football \& Basketball} = 45 \] \[ \text{Football \& Hockey} = 70 \] \[ \text{Hockey \& Basketball} = 50 \] Total viewers watching at least two games: \[ 45 + 70 + 50 = 165 \]
Step 3: Find viewers who watch exactly one game. \[ \text{Exactly one} = \text{At least one} - \text{At least two} \] \[ = 450 - 165 = 315 \]
Hence, the number of viewers who watch exactly one of the three games is \[ \boxed{315} \]