Answer
See the detailed answer below.
Work Step by Step
a) We know that $Z=6$ for carbon.
We will start with $n=1$ in the order of $(n,l,m_l,m_s)$:
$$\left(1,0,0,\frac{-1}{2}\right),
\left(1,0,0,\frac{+1}{2}\right),
\left(2,0,0,\frac{-1}{2}\right),$$
$$\left(2,0,0,\frac{+1}{2}\right),
\left(2,1,-1,\frac{-1}{2}\right),
\left(2,1,-1,\frac{+1}{2}\right)$$
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b)
We know that $Z=13$ for aluminum.
We will start with $n=1$ in the order of $(n,l,m_l,m_s)$:
$$\left(1,0,0,\frac{-1}{2}\right),
\left(1,0,0,\frac{+1}{2}\right),
\left(2,0,0,\frac{-1}{2}\right),$$
$$\left(2,0,0,\frac{+1}{2}\right),
\left(2,1,-1,\frac{-1}{2}\right),
\left(2,1,-1,\frac{+1}{2}\right),$$
$$\left(2,1,0,\frac{-1}{2}\right),
\left(2,1,0,\frac{+1}{2}\right),
\left(2,1, 1,\frac{-1}{2}\right),$$
$$\left(2,1,1,\frac{+1}{2}\right),
\left(3,0,0,\frac{-1}{2}\right),
\left(3,0, 0,\frac{+1}{2}\right),\left(3,1, -1,\frac{-1}{2}\right)$$
