Discussion
What is the coefficient of \(x^{2}y^{2}z^{3}\) in \((x + y + z)^{7}\) ?
Answer: B
The coefficient is 210.
By the multinomial theorem the coefficient of \(x^{a}y^{b}z^{c}\) in \((x + y + z)^{n}\), with \(a + b + c = n\), is \(\dfrac{n!}{a!\,b!\,c!}\). Here \(n = 7\), \(a = 2\), \(b = 2\), \(c = 3\):
\[ \begin{aligned} \frac{7!}{2!\,2!\,3!} &= \frac{5040}{2 \cdot 2 \cdot 6} \\ &= \frac{5040}{24} = 210 \end{aligned} \]
Counting another way: choose the 3 brackets that give \(z\) (\(\binom{7}{3} = 35\)), then 2 of the remaining 4 for \(x\) (\(\binom{4}{2} = 6\)): \(35 \times 6 = 210\). 420 and 630 are this times 2 and 3, from dividing by too few factorials.
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