SOLUTION: Using the letters in "PROPOSITIONS" a) How many of the distinguishable arrangements have the two Ps adjacent and the three Os adjacent? b) How many of the distinguishable arr

Question 846461: Using the letters in "PROPOSITIONS"
a) How many of the distinguishable arrangements have the two Ps adjacent and the three Os adjacent?
b) How many of the distinguishable arrangements do not have the two Ps adjacent?
Answer by Edwin McCravy(20055) (Show Source): You can put this solution on YOUR website!

         PROPOSITIONS
it has 2 P's, 3 O's, 2 S's, 2 I's, 1 R, 1 T, 1 N

Using the letters in "PROPOSITIONS"
a) How many of the distinguishable arrangements have the two Ps adjacent and the three O's adjacent?

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  = 90720

b) How many of the distinguishable arrangements do not have the two Ps adjacent?

First we find the number of distinguishable arrangements:

 = 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  = 1663200

So we subtract

9979200 - 1663200 = 8316000

Answer: 8316000

Edwin

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