SOLUTION: How many different arrangements can be obtained from the letters of the word ASSISTANT such that the letter N always appears before all the vowels?

Question 1205237: How many different arrangements can be obtained from the letters of the word
ASSISTANT such that the letter N always appears before all the vowels?
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!


Case 1. N comes 1st.
Then the distinguishable arrangements of ASSISTAT come after
the N.  That's  ways for N coming 1st

Case 2. N comes 2nd.
Then either an S or a T is before the N
Subcase 2a. S is before the N
So we have the distinguishable arrangements of ASISTAT after N,
which is  ways.
Subcase 2b. T is before the N
So we have the distinguishable arrangements of ASSISAT
after N, which is  ways.
So that's 630+420=1050 ways for N coming 2nd

Case 3. N comes 3rd.
Then SS, ST, TS, or TT is before N
Subcase 3a. SS is before the N
So we have the distinguishable arrangements of AISTAT
after N, which is  ways.
Subcase 3b. The arrangements of ST come before N, which can be
any of 2!=2 ways. Then we have the distinguishable arrangements of ASISAT
after N, which is  ways. So that's (2)(180)=360 ways
Subcase 3c. TT is before the N
So we have the distinguishable arrangements of ASSISA
after N, which is  ways.
That's 180+360+60=600 ways for N coming 3rd.

Case 4. N comes 4th.
Subcase 4a. SSS comes before N
So we have the distinguishable arrangements of AITAT
after N, which are  ways.
Subcase 4b. The distinguishable arrangements of STT comes before N, which
can be  ways. Then the distinguishable arrangements of ASISA
come after N, which is  ways.  That's (3)(30)=90 ways.
Subcase 4c. The distinguishable arrangement of SST comes before N, which
can be any of  ways. Then a distinguishable arrangement of AISAT
comes after N, which is  ways.  That's (3)(60)=180 ways.  
That's 30+90+180=300 ways for N coming 4th.

Case 5. N comes 5th.
Subcase 5a. The distinguishable arrangements of SSST come before N, which can be
any of ways. Then the distinguishable arrangements of AIAT come
after N, which is  ways. That's (4)(12)=48 ways.
Subcase 5b. The distinguishable arrangements of SSTT before N, which can be any
of ways. Then the distinguishable arrangements of AISA come after N, 
which is  ways. That's (6)(12)=72 ways.
So for Case 5, that's 48+72=120 ways for N coming 6th.
  
Case 6. N comes 6th.
The distinguishable arrangements of SSSTT come before N, which can be any of
 ways. Then the distinguishable arrangements of AIA come after
N, which is  ways.  That's (10)(3)=30 ways for N coming 6th.
(N cannot come any farther to the right because the 3 vowels must be right of N).

So for all 6 cases, the total is 1680+1050+600+300+120+30 = 3780.

Edwin

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