Henry Ernest Dudeney/Modern Puzzles/166 - Picture Presentation/Solution/Proof 1

Modern Puzzles by Henry Ernest Dudeney: $166$

A wealthy collector had ten valuable pictures.

He proposed to make a presentation to a public gallery, but could not make up his mind as to how many he would give.

So it amused him to work out the exact number of different ways.

You see, he could give any one picture, any two, any three, and so on, or give the whole ten.

$1023$, or $1024$ if you include the possibility of not giving any of his pictures at all.

For each picture, the collector can choose to give it away or to keep it.

He has to make this choice $10$ times in a row.

Hence he would have $2^{10} = 1024$ options, including not giving away any.

$\blacksquare$