At a small gathering every person shook hands exactly once with every other person. Altogether 45 handshakes took place. How many people were there?

With $n$ people each shakes $n-1$ hands, and every handshake is counted twice that way, so the number of handshakes is $n\times (n-1)÷2$. Setting $n\times (n-1)=90$ gives $n=10$, since $10\times 9=90$. The trap is 9: that is how many hands each person shook, not how many people came. 45 simply repeats the handshake count.