Question 1176491
ELLIPSES has {{{8}}} letters, {{{2}}} {{{Es}}}, {{{2}}}{{{Ls}}}, and {{{2}}}{{{ S}}}, {{{1}}} {{{I}}}, and {{{1}}} {{{P}}} 

The formula for indistinguishable permutation is

n!/(n[1]!n[2]!.....n[k]!)

where {{{n}}} is the total number of objects and  are the number of indistinguishable objects.

We have {{{2Es}}}, {{{2Ls}}}, and {{{2S}}} ({{{1}}} {{{I}}}, and {{{1}}} {{{P}}} will not make any difference); the formula then becomes:

{{{8!/(2!2!2!)}}}

={{{(8*7*6*5*4*3*2)/(2*2*2)}}}

={{{(8*7*cross(6)3*5*cross(4)2*3*cross(2))/(cross(2)*cross(2)*cross(2))}}}

={{{8*7*3*5*2*3}}}

={{{5040}}}

There are {{{5040}}} distinguishable permutations of the word {{{ELLIPSES}}}.