Number of Permutations of $$n$$ Distinct Objects

The letters of $$\text{MATRIX}$$ in alphabetical order are: $$A,I,M,R,T,X$$

Each starting letter contributes $$5!=120$$ words.

Ranks $$1\text{ to }120$$ start with $$A,$$

ranks $$121\text{ to }240$$ start with $$I,$$ and 

ranks $$241\text{ to }360$$ start with $$M.$$

Therefore,

$$W_1$$ starts with $$M.$$

Now fix $$M$$ and examine the second letter.

Each choice contributes $$4!=24$$ words.

$$MA:\ 241\text{ to }264$$

$$MI:\ 265\text{ to }288$$

$$MR:\ 289\text{ to }312$$

Hence,

$$313\text{ onwards}$$ begin with $$MT.$$

Now arrange the remaining letters $$A,I,R,X$$ in dictionary order: 

1. $$MTAIRX\quad (313)$$

2. $$MTAIXR\quad (314)$$

Therefore,

$$W_1=\text{MTAIXR}$$

Arrange the first three letters $$M,T,A$$ alphabetically: $$A,M,T$$

Keeping the last three letters unchanged, $$W_2=\text{AMTIXR}$$

Now find the rank of $$\text{AMTIXR}.$$

Words beginning with $$A$$ come first.

Fix $$A.$$

Now consider the second letter: $$AI:\ 4!=24$$

Hence,

$$25^\text{th}$$ word onwards begin with $$AM.$$

Fix $$AM.$$

Now consider the third letter:

$$AMI:\ 3!=6$$

$$AMR:\ 3!=6$$

Therefore,

$$37^\text{th}$$ word onwards begin with $$AMT.$$

Fix $$AMT.$$

Remaining letters are $$I,R,X.$$

Their dictionary order starts as:

1. $$AMTIRX\quad (37)$$

2. $$AMTIXR\quad (38)$$

Hence, the rank of $$W_2$$ is

$$\boxed{38}$$