Question 846461
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         PROPOSITIONS
it has 2 P's, 3 O's, 2 S's, 2 I's, 1 R, 1 T, 1 N
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Using the letters in "PROPOSITIONS"

a) How many of the distinguishable arrangements have the two Ps adjacent and the three O's adjacent?
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That's the distinguishable arrangements of these 9 things
{PP,OOO,R,S,S,I,I,T,N}

The two S's are indistinguishable and the two I's are indistinguishable:

That's {{{9!/(2!2!)}}} = 90720
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b) How many of the distinguishable arrangements do not have the two Ps adjacent?
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First we find the number of distinguishable arrangements:

{{{12!/(2!3!2!2!)}}} = 9979200

Next we find the number that have the 2 P's adjacent.

That is the number of distinguishable arrangements of these 11 things

{PP,O,O,O,S,S,I,I,R,T,N}

The 3 O's, the 2 S's and the 2 I's are indistinguishable:

That's {{{11!/(3!2!2!)}}} = 1663200

So we subtract

9979200 - 1663200 = 8316000

Answer: 8316000

Edwin</pre>