Question 1197425
i) If each letter must be used


"PEPSUDENT" has {{{9}}} letters, {{{2}}} P, {{{2}}} E, {{{1}}} S, {{{1 }}}U,{{{ 1 }}}D, {{{1}}} N, and {{{1}}}T

 there are {{{7}}} distinct letters to use, and  if each letter must be used ones 

that is "{{{7}}} Position {{{7}}}" or 

{{{7P7 = 7*6*5*4*3*2*1 = 5040}}} ways  

Here's why that formula works:

Choose the 1st letter any of the {{{7 }}}ways.
Choose the 2nd letter any of the remaining {{{6 }}}ways.
Choose the 3rd letter any of the remaining {{{5}}} ways.
Choose the 4th letter any of the remaining {{{4 }}}ways.
Choose the 5th letter any of the remaining {{{3}}} ways.
Choose the 6th letter either of the remaining {{{2}}} ways.
Choose the 7th letter only the remaining {{{1}}} way.

That's {{{7P7 = 7*6*5*4*3*2*1 = 7! = 5040}}}



ii) If some or all the letters may be omitted.

That's 

{{{7P0+7P1+7P2+7P3+7P4+7P5+7P6+7P7}}} 
or
       {{{1+7+42+210+840+2520+5040+5040 = 13699}}}