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If ‘S’ is the sum of the coefficients in the expansion of ${(p x + q y + r z)}^{n}$ , p, q, r > 0, then $\lim_{n \to \infty} \frac{S}{{(S^{\frac{1}{n}} + 1)}^{n}} =$

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$e^{- \frac{p q}{r}}$

$\frac{p q}{r}$

$e^{\frac{p q}{r}}$

answer is C.

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Detailed Solution

$S = {(p + q + r)}^{n} \Rightarrow \frac{S}{S^{\frac{1}{n}} + 1} = {(\frac{p + q + r}{p + q + r + 1})}^{n}$
$\lim_{n \to \infty} \frac{S}{{(S^{\frac{1}{n}} + 1)}^{n}} = 0 [∵ \frac{p + q + r}{p + q + r + 1} < 1]$

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