# Thread: Different ways 5 switches can be on or off

1. ## Different ways 5 switches can be on or off

I play a game called Castlevania HD on Xbox LIVE. When playing with other people, one signifies he or she is ready to start the match by choosing the "Ready" option (see attached image), causing an icon on his or her playercard to change from a cross against a red background (not ready) to a circle against a green background (ready). In other words, there are two possible statuses per person (ready or not ready, on or off).

In a lobby with 4 other people (for a total of 5 people in the lobby), I realized the series of ready and not ready statuses was a code (a binary code to be precise). Then, I calculated there were $\displaystyle 2^5 = 32$ possible messages with 5 people in a lobby.

I informed the lobby of my realization (withholding my calculation). Afterward, I asked the lobby how many possible messages were there with 5 people in the lobby. Someone answered there were 120 possible messages. When I asked him how he arrived at his answer, he replied with $\displaystyle 5 \times 4 \times 3 \times 2 \times 1$ (or $\displaystyle 5!$). When I told him there were 2 possible statuses for each of the 5 people and therefore $\displaystyle 2^5 = 32$ possible messages, he changed his answer to $\displaystyle 2 \times 120 = 240$.

I used an obvious example to point out the flaw in his formula: the case in which 1 person is in the lobby. (His formula offers $\displaystyle 1! = 1$ while there are 2 possible messages with 1 person in a lobby, on or off.) He responded by saying his formula isn't always valid, but I don't think the formula changes depending on the number of people in the lobby.

As a student of probability and statistics, I would like to know whose reasoning is correct. How many possible messages are there with 5 people and 2 possible statuses per person? Thank you in advance.

2. Your reasoning is correct if you don't take into account the order of the players' names (i.e. the order in which the players enter the lobby?). With his reasoning you count the number of possible orderings of the players.

3. Actually it's 31.