Prove that this alternating sum equals the factorial of x. [closed]

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While studying difference tables, I discovered this nice little formula for the factorial of a number:



$$
x! = sum_n=0^x binomxn (x-n+1)^x(-1)^n
$$



I discovered this formula empirically, but I would really like to see a proof that this equivalence holds.




calculus combinatorics summation factorial finite-differences




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closed as unclear what you're asking by Arnaud Mortier, amWhy, Xander Henderson, choco_addicted, max_zorn Aug 1 at 5:52


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.














  • Rephrased now! @ArnaudMortier
    – Blake Steines
    Jul 31 at 23:41










  • Yes! I believe the factorial is only defined for non-negative integers... otherwise we would need the gamma function.
    – Blake Steines
    Jul 31 at 23:48










  • Looks like it might be an inclusion-exclusion formula.
    – Thomas Andrews
    Jul 31 at 23:48






  • 2




    Duplicate of this: math.stackexchange.com/questions/2765969/…
    – Arnaud Mortier
    Jul 31 at 23:51










  • As per the duplicate above, the $x+1$ in the middle bracket could be replaced by any number (in fact, it could be replaced by an abstract variable).
    – Arnaud Mortier
    Jul 31 at 23:52














up vote
0
down vote

favorite
1












While studying difference tables, I discovered this nice little formula for the factorial of a number:



$$
x! = sum_n=0^x binomxn (x-n+1)^x(-1)^n
$$



I discovered this formula empirically, but I would really like to see a proof that this equivalence holds.




calculus combinatorics summation factorial finite-differences




share|cite|improve this question













closed as unclear what you're asking by Arnaud Mortier, amWhy, Xander Henderson, choco_addicted, max_zorn Aug 1 at 5:52


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.














  • Rephrased now! @ArnaudMortier
    – Blake Steines
    Jul 31 at 23:41










  • Yes! I believe the factorial is only defined for non-negative integers... otherwise we would need the gamma function.
    – Blake Steines
    Jul 31 at 23:48










  • Looks like it might be an inclusion-exclusion formula.
    – Thomas Andrews
    Jul 31 at 23:48






  • 2




    Duplicate of this: math.stackexchange.com/questions/2765969/…
    – Arnaud Mortier
    Jul 31 at 23:51










  • As per the duplicate above, the $x+1$ in the middle bracket could be replaced by any number (in fact, it could be replaced by an abstract variable).
    – Arnaud Mortier
    Jul 31 at 23:52












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





While studying difference tables, I discovered this nice little formula for the factorial of a number:



$$
x! = sum_n=0^x binomxn (x-n+1)^x(-1)^n
$$



I discovered this formula empirically, but I would really like to see a proof that this equivalence holds.




calculus combinatorics summation factorial finite-differences




share|cite|improve this question













While studying difference tables, I discovered this nice little formula for the factorial of a number:



$$
x! = sum_n=0^x binomxn (x-n+1)^x(-1)^n
$$



I discovered this formula empirically, but I would really like to see a proof that this equivalence holds.





calculus combinatorics summation factorial finite-differences





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 31 at 23:48









Brian Tung

25.1k32353




25.1k32353









asked Jul 31 at 23:26









Blake Steines

574




574




closed as unclear what you're asking by Arnaud Mortier, amWhy, Xander Henderson, choco_addicted, max_zorn Aug 1 at 5:52


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.






closed as unclear what you're asking by Arnaud Mortier, amWhy, Xander Henderson, choco_addicted, max_zorn Aug 1 at 5:52


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.













  • Rephrased now! @ArnaudMortier
    – Blake Steines
    Jul 31 at 23:41










  • Yes! I believe the factorial is only defined for non-negative integers... otherwise we would need the gamma function.
    – Blake Steines
    Jul 31 at 23:48










  • Looks like it might be an inclusion-exclusion formula.
    – Thomas Andrews
    Jul 31 at 23:48






  • 2




    Duplicate of this: math.stackexchange.com/questions/2765969/…
    – Arnaud Mortier
    Jul 31 at 23:51










  • As per the duplicate above, the $x+1$ in the middle bracket could be replaced by any number (in fact, it could be replaced by an abstract variable).
    – Arnaud Mortier
    Jul 31 at 23:52
















  • Rephrased now! @ArnaudMortier
    – Blake Steines
    Jul 31 at 23:41










  • Yes! I believe the factorial is only defined for non-negative integers... otherwise we would need the gamma function.
    – Blake Steines
    Jul 31 at 23:48










  • Looks like it might be an inclusion-exclusion formula.
    – Thomas Andrews
    Jul 31 at 23:48






  • 2




    Duplicate of this: math.stackexchange.com/questions/2765969/…
    – Arnaud Mortier
    Jul 31 at 23:51










  • As per the duplicate above, the $x+1$ in the middle bracket could be replaced by any number (in fact, it could be replaced by an abstract variable).
    – Arnaud Mortier
    Jul 31 at 23:52















Rephrased now! @ArnaudMortier
– Blake Steines
Jul 31 at 23:41




Rephrased now! @ArnaudMortier
– Blake Steines
Jul 31 at 23:41












Yes! I believe the factorial is only defined for non-negative integers... otherwise we would need the gamma function.
– Blake Steines
Jul 31 at 23:48




Yes! I believe the factorial is only defined for non-negative integers... otherwise we would need the gamma function.
– Blake Steines
Jul 31 at 23:48












Looks like it might be an inclusion-exclusion formula.
– Thomas Andrews
Jul 31 at 23:48




Looks like it might be an inclusion-exclusion formula.
– Thomas Andrews
Jul 31 at 23:48




2




2




Duplicate of this: math.stackexchange.com/questions/2765969/…
– Arnaud Mortier
Jul 31 at 23:51




Duplicate of this: math.stackexchange.com/questions/2765969/…
– Arnaud Mortier
Jul 31 at 23:51












As per the duplicate above, the $x+1$ in the middle bracket could be replaced by any number (in fact, it could be replaced by an abstract variable).
– Arnaud Mortier
Jul 31 at 23:52




As per the duplicate above, the $x+1$ in the middle bracket could be replaced by any number (in fact, it could be replaced by an abstract variable).
– Arnaud Mortier
Jul 31 at 23:52















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