The given word is MONDAY

Arranging the letters alphabetically, we get

ADMNOY

When the word starts with any of the letters A/D, the number of possibilities$=5!\times 2=240$

Now when the word starts with MA, then the number of possibilities$=4!=24$

Now when the word starts with MD, then the number of possibilities $=4!=24$

Now when the word starts with MN, then the number of possibilities $=4!=24$

Now when the word starts with MOA, then the number of possibilities$=3!=6$

Now when the word starts with MOD, then the number of possibilities $=3!=6$

Now when the word starts with MONA, then the number of possibilities $=2!=2$

Now when the word starts with MONDAY, then the number of possibilities$=1$

Rank $=240+24\times 3+6\times 2+2+1=327$

Hence, rank of the word MONDAY is $327$.

SHORTCUT METHOD

Another method is a shortcut, first number the alphabet and then check how many number are there on right hand side less that number and then start number in factorial in decreasing order and then multiply and add them and in last add one to find rank, 

$\begin{array}{cccccc}3& 5& 4& 2& 1& 6\\ M& O& N& D& A& Y\\ 2& 3& 2& 1& 0& 0\\ 5!& 4!& 3!& 2!& 1!& 0!\end{array}$

$\therefore$ Rank $=(2\times 5!+3\times 4!+2\times 3!+1\times 2!)$

$=240+72+12+2+1=327$