Binomial Expansion

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

This hard difficulty mathematics question is from the chapter binomial theorem and its simple applications, covering the topic of binomial expansion. It appeared in the 2025 exam.

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Binomial Expansion

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