How do you use the binomial formula to find the coefficient of the $z^{19} q^{2}$ $z^19 q^2$ term in the expansion of ${(z + 2 q)}^{21}$ $(z+2q)^21$ ?

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Answer 1 · 2016-01-19T22:17:26

$210$ $210$

The coefficient will be the second term in the twenty-first row of Pascal's triangle (excluding the inside and outside ones).

This can be represented as $_{21} C_{2}$ $""_21C_2$ , or $21$ $21$ choose $2$ $2$ .

Algebraically represented, this is

$\frac{21!}{(2!) (19!)} = \frac{21 \times 20}{2} = 210$ $(21!)/((2!)(19!))=(21xx20)/2=210$

How do you use the binomial formula to find the coefficient of the z19q2z^19 q^2 term in the expansion of (z+2q)21(z+2q)^21?