Hello, Judi!
Code:
1 2 3
N R
+ R N
-----
A B C
The addition problem above is correct.
If N, R, A, B, and C are different digits,
what is the greatest possible value of $\displaystyle B+C$?
In column-1, we see that $\displaystyle A = 1.$ Code:
1 2 3
N R
+ R N
-----
1 B C
In column-3, we see that $\displaystyle R + N$ ends in $\displaystyle C.$
In column-2, we see that $\displaystyle N + R$ ends in $\displaystyle B.$
Then $\displaystyle R + N \geq 10$ and there is a "carry" to column-2
. . where we have: $\displaystyle N + R + 1 \:=\:10 + B$
For maximum $\displaystyle B + C$, let $\displaystyle \{R,N\} = \{8,9\}$
. . But we find that this results in duplicated digits.
For $\displaystyle R = 9$ we have: Code:
N 9
+ 9 N
-----
A B C
But we get: $\displaystyle B = N.$
. . (If $\displaystyle N = 9$, then $\displaystyle B = R.$)
Hence, neither $\displaystyle R$ nor $\displaystyle N$ can be $\displaystyle 9.$
The next largest sum occurs for: $\displaystyle \{R,N\} = \{7,8\}$ and we have: Code:
8 7
+ 7 8
-----
1 6 5
Therefore: $\displaystyle B + C \,= \,11$