Why 0! Equals 1
0! equals 1 because factorials count arrangements, and there is exactly one way to arrange zero objects.
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Why is 0! defined as 1 instead of 0 or undefined?
You know that 1! = 1, 2! = 2, 3! = 6 — but what about 0!?
Factorials mean 'multiply downward from n,' and there is nothing to multiply when n = 0. So why do mathematicians insist that 0! = 1?
Show how the factorial pattern 5!/5 = 4!, 4!/4 = 3! keeps extending downward, forcing an empty product to equal 1.
0! = 1 because factorial is defined so that removing a factor leaves the smaller factorial — the same rule that makes multiplying no numbers yield 1.
- Gamma function
- permutation formulas for n > 0
- combinatorial proofs beyond the empty-product argument
- 01The Missing FactorialslideSlot 1Hook
Present a short list of factorial values (1!, 2!, 3!, 4!) and stop at the question: what comes before 1!?
- Factorials shrink by dividing by n
- The pattern n! = n × (n−1)!
- What value keeps the chain unbroken?
PhenomenonThe familiar factorial sequence 1, 2, 6, 24 ends with 1! at the top, leaving a hole where 0! should be.
QuestionWhat number must 0! be so that dividing 1! by 1 still gives the next factorial down?
- 02Two Competing IntuitionsslideSlot 2Tension
Contrast the 'nothing was multiplied, so it should be 0' intuition with the combinatorial 'arrange zero objects' intuition.
- Empty product feels like 0
- Empty set feels like one arrangement
- Pattern-consistency suggests 1
PredictionBefore seeing the answer, learners often guess 0 or 'undefined.'
Tempting intuitionIt feels natural to say that multiplying no numbers together gives 0, just as adding nothing gives 0.
- 03The Empty Productinteractive
Show the staircase 4!/4 = 3!, 3!/3 = 2!, 2!/2 = 1!, 1!/1 = 0!, forcing 0! = 1 to keep the recurrence alive.
- n! = n × (n−1)! implies (n−1)! = n!/n
- Following the chain gives 0! = 1!/1 = 1
- Arranging zero items yields exactly one arrangement
- 04The Same Trick in a New PlaceslideSlot 4Takeaway
Transfer the empty-product idea to a nearby situation: the number of ways to choose 0 items from n.
- nC0 = 1 by the same identity
- Binomial coefficients stay consistent
- Empty operations in math usually default to the identity element
TransferApply the same reasoning to combinations: nC0 asks how many ways to choose nothing from n items, and the formula nC0 = 1 mirrors 0! = 1.
Expected inferenceLearners should expect that any 'empty' operation — choosing nothing, multiplying nothing, intersecting nothing — defaults to the identity element of that operation.
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