For any base $b$, if we add $a$ and $b$ with $k$ carries, then $S_b(a+b)=S_b(a)+S_b(b)-(b-1)k$, where $S_b$ denotes the sum of digits. Since the resulting sum is independent of the order of addition, the total number of carries is independent as well.

For any base $b$, if we add $a$ and $c$ with $k$ carries, then $S_b(a+c)=S_b(a)+S_b(c)-(b-1)k$, where $S_b$ denotes the sum of digits. Since the resulting sum is independent of the order of addition, the total number of carries is independent as well.

For any base $b$, if we add $a$ and $b$ with $k$ carries, then $S_b(a+b)=S_b(a)+S_b(b)-(b-1)k$, where $S_b$ is the sum of digits. Since the resulting sum is independent of the order of addition, the total number of carries is independent as well.

For any base $b$, if we add $a$ and $b$ with $k$ carries, then $S_b(a+b)=S_b(a)+S_b(b)-(b-1)k$, where $S_b$ denotes the sum of digits. Since the resulting sum is independent of the order of addition, the total number of carries is independent as well.