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## Main Question or Discussion Point

Suppose f(n,p)=integral(n!/(x! (n - x)!dx, for x from -1/2 to p)

where n>1, p<n+1/2

Are approximate formulas known for this kind of integral?

Empirically, f(n,n+1/2) seems to be close to 2^n

More generally, I'm looking for approximate formulas for integrals of n!/(x1!x2!...xn!) over nice sets, textbook suggestions are welcome

ListPlot[Table[2^n - NIntegrate[n!/(x! (n - x)!), {x, -1/2, n + 1/2}], {n, Range[30]}], PlotRange -> {{0, 30}, {0, .12}}]

where n>1, p<n+1/2

Are approximate formulas known for this kind of integral?

Empirically, f(n,n+1/2) seems to be close to 2^n

More generally, I'm looking for approximate formulas for integrals of n!/(x1!x2!...xn!) over nice sets, textbook suggestions are welcome

ListPlot[Table[2^n - NIntegrate[n!/(x! (n - x)!), {x, -1/2, n + 1/2}], {n, Range[30]}], PlotRange -> {{0, 30}, {0, .12}}]