quantum mechanics - Can we use the Pascal triangle as an aid to construct superpositions of wavefunctions corresponding to $n$ electron spins?

Suppose we have n electrons and want to construct the wavefunction corresponding to the spins of the electrons. Can we construct this wavefunction (in the $(s_1,s_2,s_3 ... s_n)$ representation, so not the $(S^2,S_z)$ representation; the C.G. coefficients relate the two representations) in terms of Dirac's kets (every arrow in these kets corresponds to a Dirac spinor $\left(\begin{array}{c} 1\\0\\\end{array}\right)$ for one spin up, and $\left(\begin{array}{c} 0\\1\\\end{array}\right)$ for one spin down) by using the Pascal Triangle (PT)?

Let me explain. The 1 in row 0 of the PT corresponds to a single electron in either a spin up or a spin down ket:

$$|\uparrow\rangle {or} |\downarrow\rangle$$

Obviously, here is no superposition invovled.

The superposition of the two spins of one electron spin is composed of 1 spin-up electron and 1 spin-down electron, corresponding to the two 1's in row 1 in the PT. Because the wavefunction has to be normalized and be antisymmetric we'll get the following result:

$$+\frac{1}{\sqrt{2}}|\uparrow\rangle-\frac{1}{\sqrt{2}}|\downarrow\rangle$$

For two electron spins, we have 1 time a spin-up,spin-up ket, 2 times a spin-up, spin-down ket, and one time a spin-down, spin-down ket (corresponding to the 1,2,1 in the second row in the PT). Again, taking normalization and antisymmetry into account [which simply can be done by putting a factor $\frac{1}{\sqrt{n}}$ in front of the wavefunction and distribute the +'s and -'s in a symmetric way: interchanging opposite kets of the total wavefunction, which is obvious in the case of the wavefunction for one electron spin in a superposition state involving a ket for spin up and a ket for spin down (the one above) will yield the negative of the original] we'll get:

$$\frac{1}{\sqrt{4}}(+|\uparrow\uparrow\rangle+|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle-|\downarrow\downarrow\rangle)$$

So for a superposition of three electron spins, we'll get 1 ket with three spins up, 3 with two up and one down, 3 with one up and two down, and 1 with three spins down, the 1,3,3, and 1 corresponding to the 1,3,3,1, in row 3 in the PT. We get:

$$\frac 1 {\sqrt{8}}(+|\uparrow\uparrow\uparrow\rangle+|\uparrow\uparrow\downarrow\rangle+|\uparrow\downarrow\uparrow\rangle+|\downarrow\uparrow\uparrow\rangle-|\uparrow\downarrow\downarrow\rangle-|\downarrow\uparrow\downarrow\rangle-|\downarrow\downarrow\uparrow\rangle-|\downarrow\downarrow\downarrow\rangle)$$

For a superposition of four electron spins we'll get the number of four spins up (or down) by looking to the left and right of the PT, which is always 1, so there is 1 ket with all spins up and one with four spins down (and a minus sign). There are 4 kets with three spins up, one spin down (and a plus sign), 6 with two up, two down (of which 3 have a plus sign and 3 a minus sign; see the two electron spin superposition above), and 4 with one up, three down (and a minus sign). The normalization factor is $\frac{1}{\sqrt{16}}$. I won't elaborate to write the complete function. The 1,4,6,4,and 1 correspond to the 1,4,6,4,1 in row 4 of the PT.

For a superposition of five electron spins we can use the numbers of row five in the PT to know the distributions of the different kets: 1 up,up,up,up,up, 5 up,up,up,up,down, 10 up,up,up,down,down, 10 up,up,down,down,down, 5 up,down, down,down,down, 1 down,down,down,down,down, after which we add them antisymmetrically, and put the normalization factor in front.

For small numbers of electron spins this is rather trivial, but it comes in handy when the number increases. Can it be done like this?