1. In how many ways can 36 identical robots be assigned to five assembly lines with

(a) at least four robots assigned to each line? (5%)
(b) at least four, but no more than ten, assigned to each line? (5%)

(a) $\binom{20}{4}$
(b) $\binom{20}{16} – \binom{5}{1}\binom{13}{9} + \binom{5}{2}\binom{6}{2}$

2. Let $D = \begin{bmatrix} 2 & -1\\ -1 & 2 & -1 & & 0\\ & -1 & 2 & \ddots\\ & & -1 & \ddots & -1\\ & 0 & & \ddots & 2 & -1\\ & & & & -1 & 2 \end{bmatrix} , \, i.e., \, \begin{equation} D(i,j) = \begin{cases} 2, & \text{if$i = j.$}\\ -1, & \text{if$|i – j| = 1.$}\\ 0, & \text{elsewhere.} \end{cases} \end{equation}$

(a) Use recurrence relation to express determinant of $D, \, i.e., \, |D|.$ (5%)
(b) Find the general solution for $|D_n|.$ (5%)
(c) $|D_1| = 2, |D_2| = \begin{vmatrix} 2 & -1\\ 1 & 2 \end{vmatrix} = 3.$ Find $|D_{100}|.$ (5%)

(a) $D_n = 2D_{n-1} – D_{n-2}, \, \forall n \geq 3. \ D_1 = 2, \, D_2 = 3.$
(b) $|D_n| = n + 1$
(c) $|D_{100}| = 101$

3. Solve the following recurrence relations: $6a_n – 5a_{n-1} + a_{n-2} = \sin(n \pi)$ with $a_0 = 1, \, a_{-1} = a_{-2} = 0.$ (10%)

4. Let $(Q,\oplus,\otimes)$ denote the field, where $\oplus$ and $\otimes$ are defined by $a \oplus b = a + b – k, \, a \otimes b = a + b – (ab/m),$ for fixed elements $k,m (\neq 0)$ of $Q.$ Determine the values for $k$ and $m$ in each of the following:

(a) The zero element for the field is 5. (5%)
(b) The additive inverse of the element 8 is -7. (5%)
(c) The multiplicative inverse of 3 is 1/6. (5%)

(a) $k = 5, \, m = 5.$
(b) $k = \dfrac{1}{2}, \, m = \dfrac{1}{2}.$
(c) 略