Wikipedia:Reference desk/Archives/Mathematics/2016 March 31

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March 31

Not quite a permutation

I have three lightbulb a in a row. Each bulb can be either on or off. I want to work out how many possible orders there are, ie. if we call 'on' and 'off' light and dark: LLL LLD LDD DDD DDL ...etc.

I think there are 8. But what is this called? It's not quite either combinations or permutations. What's the formula for larger sets? Amisom (talk) 02:28, 31 March 2016 (UTC)

See Power set#Properties. -- ToE 02:49, 31 March 2016 (UTC)

Also related are the Binomial theorem and, more generally, the Multinomial theorem. Given n ordered lightbulb which can be in two states, there are 2ⁿ possible arrangements. If they have three possible states (high, low, & off), then there are 3ⁿ possible arrangements.

As for the combinatorics term, I don't know if there is anything more specific than the Rule of product.

(Number of allowable states of item one) · (Number of allowable states of item two) · ... · (Number of allowable states of item n).

So if you have 3 two-state bulbs (on/off) and 2 three-state bulbs (high/low/off), the total number of possible states are 2·2·2·3·3 = 2³·3² = 8·9 = 72 possible arrangements. -- ToE 03:39, 31 March 2016 (UTC)

This is the fundamental counting principle, no? Joseph A. Spadaro (talk) 04:53, 7 April 2016 (UTC)