Problems

Solution:

1. For each number from \(1\) to \(8\) (4+4), we can count the number of ways it can be achieved in any way, or without including a leading \(0\). \begin{array}{c|c|c|c} \text{total} & \text{ways with }0 & \text{ways without } 0 & \text{comb}\\ \hline 8 & 1 & 1 & 1\\ 7 & 2 & 2 & 4 \\ 6 & 3 & 3 & 9 \\ 5 & 4 & 4 & 16 \\ 4 & 5 & 4 & 20 \\ 3 & 4 & 3 & 12 \\ 2 & 3 & 2 & 6 \\ 1 & 2 & 1 & 2 \\ \hline &&& 70 \end{array}
2. For \(2k\) to \(k+1\) there are \(1 \times 1 + 2 \times 2 + \cdots i\times i+\cdots + k\times k\) ways to achieve this, (we can choose anything from \(k\) to \(k-i+1\) for the first digit, and we can never have a \(0\). For \(1\) to \(k\) we can have \(1 \times 2 + 2 \times 3 + \cdots + k \times (k+1)\) since we cannot start with a \(0\), but can have anything less than or equal to \(i\) for the first digit, and then with the \(0\) we can have the same thing starting with \(0\). Hence the answer is: \begin{align*} && S &= \sum_{i=1}^k i^2 + \sum_{i=1}^k i (i+1) \\ &&&= 2\sum_{i=1}^k i^2 + \sum_{i=1}^k i \\ &&&= \frac{1}{3} k(k+1)(2k+1) + \frac12k(k+1) \\ &&&= k(k+1) \left (\frac{2k+1}{3} + \frac{1}{2} \right) \\ &&&= \frac16 k(k+1)(4k+2+3) \\ &&&= \frac16 k(k+1)(4k+5) \end{align*}