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?

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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) About Me  (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 8%21%2F%282%213%212%21%29=1680 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 7%21%2F%282%212%212%21%29=630 ways.
Subcase 2b. T is before the N
So we have the distinguishable arrangements of ASSISAT
after N, which is 7%21%2F%282%213%21%29=420 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 6%21%2F%282%212%21%29=180 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 6%21%2F%282%212%21%29=180 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 6%21%2F%282%213%21%29=60 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 5%21%2F%282%212%21%29=30 ways.
Subcase 4b. The distinguishable arrangements of STT comes before N, which
can be 3%21%2F2%21=3 ways. Then the distinguishable arrangements of ASISA
come after N, which is 5%21%2F%282%212%21%29=30 ways.  That's (3)(30)=90 ways.
Subcase 4c. The distinguishable arrangement of SST comes before N, which
can be any of 3%21%2F2%21=3 ways. Then a distinguishable arrangement of AISAT
comes after N, which is 5%21%2F%282%21%29=60 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 4%21%2F3%21=4ways. Then the distinguishable arrangements of AIAT come
after N, which is 4%21%2F2%21=12 ways. That's (4)(12)=48 ways.
Subcase 5b. The distinguishable arrangements of SSTT before N, which can be any
of 4%21%2F%282%212%21%29=6ways. Then the distinguishable arrangements of AISA come after N, 
which is 4%21%2F2%21=12 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
5%21%2F%283%212%21%29=10 ways. Then the distinguishable arrangements of AIA come after
N, which is 3%21%2F2%21=3 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