Exponent patterns $(a, b, c)$ with $a + b + c = 4$ , coefficient $\frac{4 !}{a ! b ! c !}$ :

$(4, 0, 0)$ : $1$ → $x^{4}, y^{4}, z^{4}$
$(3, 1, 0)$ : $\frac{24}{6} = 4$ → $x^{3} y, x^{3} z, y^{3} x, y^{3} z, z^{3} x, z^{3} y$
$(2, 2, 0)$ : $\frac{24}{4} = 6$ → $x^{2} y^{2}, x^{2} z^{2}, y^{2} z^{2}$
$(2, 1, 1)$ : $\frac{24}{2} = 12$ → $x^{2} y z, x y^{2} z, x y z^{2}$

$(x + y + z)^{4} = x^{4} + y^{4} + z^{4} + 4 (x^{3} y + x^{3} z + x y^{3} + y^{3} z + x z^{3} + y z^{3}) + 6 (x^{2} y^{2} + x^{2} z^{2} + y^{2} z^{2}) + 12 (x^{2} y z + x y^{2} z + x y z^{2}) .$
Check: 15 terms $= (2 4 + 2)$ , and the coefficients sum to $3 + 24 + 18 + 36 = 81 = 3^{4}$ . ✓

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