Rank of the word AUGUST in the dictionary of words formed by using the letters of the word AUGUST

if we arrange AUGUST in alphabatical order then we get,AGSTUU.so,first letter is A which we get in 0 ways.the 2nd letter U that we get 3*4!/2! ways.the third letter is G that we get 0 ways.the 3th letter U that we get in 2*2! ways.and S that we get 0 ways and T is gotten in 1 ways.so.the rank is:0+3*4!/2!+0+2*2!+0+1=41

letters according to dictionary is A G S T U U
so A is in correct order then G will come that is not in correct order so total number of words in which G comes second from right = 12
same for S, T
then U will come and it is in correct place then G comes and G is also in correct place
here we reached on three places
for fourth place S will come but its not correct place for S. so total number of words in which S comes in fourth place = 2
same as T= 2
and then U comes which is in correct place and then S and T comes .. and they are also in their respective place
so the number of word before AUGUST is= 12+12+12+2+2 = 40
so AUGUST will have 41 place in dictionary and option 1 is correct
so the

Words starting with AG XXXX = 4!/2 = 12
Words starting with AS XXXX = 4!/2 = 12
Words starting with AT XXXX = 4!/2 = 12
Words starting with AUG XXXX = AUG STU, AUGSUT, AUGTSU,AUGTUS, AUGUST

so rank of AUGUST = 12*3+5= 41

follow the sequence AGSTUU
A[g]=4!
A[s]=4!
A[t]=4!
AU--- fix
AUG--- fix
AUGS=2!
AUGT=2!
AUGU---fix
AUGUS---fix
AUGUST=1
=== === ====
3*(4!)+2*(2!)+1=41.

b'coz adding all the positions of letters in the alphabets we get this answer....

1).firstly,arrange the letter of august in alphabetical order.
2).now ,it become AGSTUU.
3).pick up first letter A.now compare A with AUGUST
this is the letter we want.so,fix it.
4).now go on search until we get second letter of AUGUST.
5).So,we have [A]G,[A]S,[A]T.DO these r the letters we want,answer is no.so,write down how many letter can we write down with above ,
so,for [A]G,[A]S,[A]T SEPARATELY,WE HAVE
4!/2!=12
SO,TOTAL=3*12=36
6).NOW ,we have find [A]U,so fix it.
7).AGAIN,start from first until u get [AU]G.if find ,fix it.
8).repeat step 5,for [AUG]_.
9).so ,we have
[AUG]S->2!,[AUG]T->2!
10).NOW, WE HAVE FIND[AUG]U.SO,FIX IT.
11).AGAIN,REPEAT STEP 5.SO ,WE HAVE[AUGU]S.
12).ATLAST,WE HAVE [AUGUS]T.SO NO.OF WORDS=1.
13).SO,IN TOTAL TO MAKE AUGUST IN ALPHABETICAL ORDER,WE HAVE
36+2+2+1=41

because, alphabetically A,G,S,T,U,U,
SO KEPPING SEQUENCIALLY G, THEN S THEN T THEN U AFTER A AND FINDING NO. OF PERMUTATION IT COMES TO 41