Binomial Expansion

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The ratio of the coefficient of x¹⁵ to the term independent of x in (x² + 2/x)¹⁵ is

RankGuru · Accepted Answer

1. **General Term**: $T_{r+1} = \binom{15}{r} (x^2)^{15-r} (\frac{2}{x})^r = \binom{15}{r} (2)^r x^{30-3r}$.
2. **Finding $x^{15}$**: $30-3r = 15 \implies 3r = 15 \implies r = 5$.
   - Coeff $= \binom{15}{5} 2^5$.
3. **Finding Independent Term**: $30-3r = 0 \implies r = 10$.
   - Term $= \binom{15}{10} 2^{10}$.
4. **Ratio Calculation**: $	ext{Ratio} = \frac{\binom{15}{5} 2^5}{\binom{15}{10} 2^{10}}$.
   - Note that $\binom{15}{5} = \binom{15}{10}$.
   - Ratio $= \frac{2^5}{2^{10}} = \frac{1}{2^5} = \frac{1}{32} \implies 1:32$.

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