# Trying to find the rank of the word permutation .

What is the rank of the word $PERMUTATION$ if all the words formed by the letters of "$PERMUTATION$" are arranged in ascending order ?

$PERMUTATION$ in ascending order in $\{A,E,I,M,N,O,P,R,T,T,U\}$

What I tried :

$(6*10!/2!) + (1*9!/2!) + (5*8!/2!) + (2*7!/2!)$ + I'm stuck here .

FYI : Answer is $11,176,100$

• I have corrected my answer below so I have the same answer as you. – Noble Mushtak Jun 19 '16 at 14:10

I am going to follow this algorithm to find the rank. Based on what you have so far, you have already gotten through the first four letters "PERM", so we continue from there.

1. Fix the "PERMA" to get $\frac{6!}{2!}$.
2. Fix the "PERMI" to get $\frac{6!}{2!}$.
3. Fix the "PERMN" to get $\frac{6!}{2!}$.
4. Fix the "PERMO" to get $\frac{6!}{2!}$.
5. Fix the "PERMT" to get $\frac{6!}{2!}$.
6. Fix the "PERMT" to get $\frac{6!}{2!}$. (Second "T")
7. Fix the "PERMU", which matches the word, so we fix it permanently.
8. Fix the "PERMUA" to get $\frac{5!}{2!}$.
9. Fix the "PERMUI" to get $\frac{5!}{2!}$.
10. Fix the "PERMUN" to get $\frac{5!}{2!}$.
11. Fix the "PERMUO" to get $\frac{5!}{2!}$.
12. Fix the "PERMUT", which matches the word, so we fix it permanently.
13. Fix the "PERMUTA", which matches the word, so we fix it permanently.
14. Fix the "PERMUTAI" to get $3!$.
15. Fix the "PERMUTAN" to get $3!$.
16. Fix the "PERMUTAO" to get $3!$.
17. Fix the "PERMUTAT", which matches the word, so we fix it permanently.
18. Fix the "PERMUTATI", which matches the word, so we fix it permanently.
19. Fix the "PERMUTATIN" to get $1$.
20. Fix the "PERMUTATIO", which matches the word, so we fix it permanently.
21. Fix the "PERMUTATION", which matches the word, so we fix it permanently. Remember to count this word since it's the end, so we get $1$.

Now, our final answer is what you got plus the above, or: $$6\frac{10!}{2!}+\frac{9!}{2!}+5\frac{8!}{2!}+2\frac{7!}{2!}+6\frac{6!}{2!}+4\frac{5!}{2!}+3\cdot 3!+2\cdot 1=11,176,100$$