Problems

Solution:

1. The largest a five-digit number can have for its digit sum is \(45 = 9+9+9+9+9\). To achieve \(43\) we can either have 4 9s and a 7 or 3 9s and 2 8s. The former can be achieved in \(5\) ways and the latter can be achieved in \(\binom{5}{2} = 10\) ways. (2 places to choose to put the 2 8s). In total this is \(15\) ways.
2. To achieve \(39\) we can have: \begin{array}{c|l|c} \text{numbers} & \text{logic} & \text{count} \\ \hline 99993 & \binom{5}{1} & 5 \\ 99984 & 5 \cdot 4 & 20 \\ 99974 & 5 \cdot 4 & 20 \\ 99965 & 5 \cdot 4 & 20 \\ 99884 & \binom{5}{2} \binom{3}{2} & 30 \\ 99875 & \binom{5}{2} 3! & 60 \\ 99866 & \binom{5}{2} \binom{3}{2} & 30 \\ 98886 & 5 \cdot 4 & 20 \\ 98877 & \binom{5}{2} \binom{3}{2} & 30 \\ 88887 & \binom{5}{1} & 5 \\ \hline && 240 \end{array}