Find the total number of arrangements of the letters in the expression a³ b² c⁴ when written at full length.

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 . 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

= 124

Hence, number of ways of arranging the letters of word/expression aaabbcccc is equals to 124.