## Total Number of Arrangements of Four Letters Picked From the Word POSSESSES

Suppose 4 letters are to be picked from the word 'POSSESSES'. How many arrangements of the selected letters are possible?We can order in terms of the number of S's picked.

If no S's are picked the only way to pick four letters in POEE. There are

\[\frac{4!}{2!}=12\]

arrangements.If one S is picked then we could pick

SOEP which can be arranged in

\[4!=24\]

ways.SOEE which can be arranged in

\[\frac{4!}{2!} = 12\]

ways.SPEE which can be arranged in

\[\frac{4!}{2!} = 12\]

ways.If two S's are picked we could pick

SSOE which can be arranged in

\[\frac{4!}{2!} = 12\]

ways.SSPE which can be arranged in

\[\frac{4!}{2!} = 12\]

ways.SSPO which can be arranged in

\[\frac{4!}{2!} = 12\]

ways.SSEE which can be arranged in

\[\frac{4!}{2!2!} = 6\]

ways.If three S's are picked then we could pick

SSSE which can be arranged in

\[\frac{4!}{3!} = 4\]

ways.SSSO which can be arranged in

\[\frac{4!}{3!} = 4\]

ways.SSSP which can be arranged in

\[\frac{4!}{3!} = 4\]

ways.If four S's are picked there is only one arrangement.

Adding up all the possibilities gives

\[12+24+12+12+12+12+12+6+4+4+4+1=115\]

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