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Coefficients

The ratio of ...

Question

The ratio of the coefficient of x¹⁵ to the term independent of x in ${(x^{2} + \frac{2}{x})}^{15},$ is
(a) 12 : 32
(b) 1 : 32
(c) 32 : 12
(d) 32 : 1

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Solution

In ${(x^{2} + \frac{2}{x})}^{15}$ we have $T_{r + 1} =^{15} C_{r} {(x^{2})}^{15 - r} {(\frac{2}{x})}^{r}$
$i . e . General term is T_{r + 1} =^{15} C_{r} x^{30 - 3 r} 2^{r}$
Hence for the term independent of x,
30 – 3r = 0
i.e. r = 10
hence T₁₁ has coefficient ¹⁵C₁₀2¹⁰ ...(1)
and term with x¹⁵will have 30 – 3r = 15
i.e. 15 = 3r
i.e. r = 5
∴ coefficient will be ¹⁵C₅2⁵ ...(2)
∴ ratio of coefficient of x¹⁵ to the term independent of x will be
$\frac{{eq}^{n} (2)}{{eq}^{n} (1)} = \frac{2^{5} \times^{15} C_{5}}{2^{10} \times^{15} C_{10}} = \frac{1}{2^{5}} (∵^{15} C_{10} =^{15} C_{5 - 10} =^{15} C_{5})$
i.e. ratio will be 1 : 32
Hence, the correct answer is option B.

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