Given, 

Expression a³b²c⁴ 

i.e. in expansion aaabbcccc. 

To find : Number of expressions that can be generated by permuting the letters of given expression aaabbcccc. 

Given expression has three repeating characters a, b, and c. 

The letter a is repeated 3 times, the letter b is repeated 2 times, and the letter c is repeated 4 times. 

So, the given problem can now be rephrased as to find a total number of arrangements of 9 objects (3+2+4) of which 3 objects are of the same type, 2 objects are of another type, and 4 objects are of different type. 

Since we know, 

Permutation of n objects taking r at a time is ⁿP_r,and permutation of n objects taking all at a time is n! 

And, 

We also know,

Permutation of n objects taking all at a time having p objects of the same type, q objects of another type, r objects of another type is \(\frac{n!}{p!\times q!\times r!}\). 

i.e.,

The number of repeated objects of same type are in denominator multiplication with factorial.

The number of ways of arranging 9 objects of which 3, 2, and 4 objects are of different types is equaled to 

= \(\frac{9!}{3!\times 2!\times 4!}\) 

= 124

Hence,

Number of ways of arranging the letters of word/expression aaabbcccc is equals to 124.