Find the coefficient of $x^{2} y z$ in the expansion of $(x + y + z)^{4}$ .

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Solution

$(x + y + z)^{4} =^{4} C_{0} x^{0} (y + z)^{4} +^{4} C_{1} x (y + z)^{3} +^{4} C_{2} x^{2} (y + z)^{2} +^{4} C_{3} x^{3} (y + z) +^{4} C_{4} x^{4} (y + z)^{3}$
We are intended in term $4 C_{2} x^{2} (y + z)^{2}$
$^{4} C_{2} x^{2} (y + z)^{2} = \frac{4 \times 3}{2} x^{2} (y^{2} + z^{2} + 2 y z)$
$= 6 x^{2} y^{2} + 6 x^{2} z^{2} + 12 x^{2} y z$
coefficient of $x^{2} y z \to 12$

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