Answer
See below
Work Step by Step
Each term in the expansion $(A+B)^k$ is made up of a series of $k$ matrices, each of which is either $A$ or $B$. We get $2^k$ separate strings if we multiply the possibilities for each location in a string of length $k$. In the expansion of $(A+B)^k$, there are $2k$ different terms.
From the exercise 46, we have
$(A+B)^3\\=(A+B)(A+B)(A+B)\\=(A+B)^2(A+B)\\=(A^2+AB+BA+B^2)(A+B)\\=A^3+A^2B+ABA+AB^2+BA^2+BAB+B^2A+B^3$
Then,
$(A+B)^4\\=(A+B)(A+B)(A+B)(A+B)\\=(A+B)^3(A+B)\\=(A+B)(A^3+A^2B+ABA+AB^2+BA^2+BAB+B^2A+B^3)\\=A^4+A^3B+A^2BA+A^2B^2+ABA^2+ABAB+AB^2A+AB^3+BA^3+BA^2B+BABA+BAB^2+B^2A^2+B^2AB+B^3A+B^4$
