The question is in how many ways can 5 letters be mailed if there are 2 mail boxes available? I would say that there are 2 ways to put the first letter (either to Box 1 or Box 2) and there are also 2 ways to put the second letter, etc. so the total number of ways is $2^{5}=32$ but my textbook says the answer is $25$ which is $5^{2}$, the other way round. Who is right?

EDIT: Maybe their answer $25$ does not come from $5^{2}$ but $2^{5}-7$ and I simply forgot to subtract some extraneous ways?

  • $\begingroup$ This should be $2^5=32$, where the extraneous case come from! $\endgroup$ – user249332 Dec 14 '15 at 15:13
  • $\begingroup$ Sorry but I don't get what you mean. $\endgroup$ – Richard Smith Dec 14 '15 at 15:36
  • $\begingroup$ The answer is purely $2^5=32$, in this process, there is no over-counting. So, you need not subtract anything (unless there are some restrictions assigned, that you missed to mention). $\endgroup$ – user249332 Dec 14 '15 at 15:48
  • $\begingroup$ Ah, ok, thanks. $\endgroup$ – Richard Smith Dec 14 '15 at 16:05
  • $\begingroup$ Textbooks do have the occasional typo. Maybe someone forgot to put the "^" in "2^5." $\endgroup$ – Barry Cipra Dec 14 '15 at 17:45

The first letter can be posted in any of the $2$ post boxes. Therefore, it has $2$ choices.

Similarly, the second, the third, the fourth and the fifth letter can each be posted in any of the $2$ post boxes.

Therefore, the total number of ways the $5$ letters can be posted in $2$ boxes is $\color{red}{ 2\times 2\times 2\times 2\times 2=32}.$


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