limn→∞ ncrmnr1−mnn−r equals

limn→∞n Crmnr1−mnn−r=limn→∞ n(n−1)(n−2)…(n−r+1)r!⋅mrnr1−mn−nm−(n−r)×mn=limn→∞ n(n−1)(n−2)…(n−r+1)nrmr=limn→∞ 1−mn−nm1−rn(−m)=(1−0)(1−0)…(1−0)r!mre−m=e−mmrr!

limn→∞ ncrmnr1−mnn−r equals

limn→∞n Crmnr1−mnn−r=limn→∞ n(n−1)(n−2)…(n−r+1)r!⋅mrnr1−mn−nm−(n−r)×mn=limn→∞ n(n−1)(n−2)…(n−r+1)nrmr=limn→∞ 1−mn−nm1−rn(−m)=(1−0)(1−0)…(1−0)r!mre−m=e−mmrr!

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$lim_{n \to \infty}^{n} c_{r} {(\frac{m}{n})}^{r} {(1 - \frac{m}{n})}^{n - r} equals$

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$e^{- m} m^{r}$

$\frac{m^{r}}{r!}$

$\frac{m^{r} e^{- m}}{r!}$

$\frac{e^{- r} r^{m}}{m!}$

answer is C.

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Detailed Solution

$\begin{array}{l} {lim}_{n \to \infty}^{n} C_{r} {(\frac{m}{n})}^{r} {(1 - \frac{m}{n})}^{n - r} \\ = lim_{n \to \infty} \frac{n (n - 1) (n - 2) \dots (n - r + 1)}{r!} \cdot \frac{m^{r}}{n^{r}} {[{(1 - \frac{m}{n})}^{- \frac{n}{m}}]}^{- \frac{(n - r) \times m}{n}} \\ = lim_{n \to \infty} \frac{n (n - 1) (n - 2) \dots (n - r + 1)}{n^{r}} (m^{r}) \\ = lim_{n \to \infty} {[{(1 - \frac{m}{n})}^{- \frac{n}{m}}]}^{(1 - \frac{r}{n}) (- m)} \\ = \frac{(1 - 0) (1 - 0) \dots (1 - 0)}{r!} m^{r} e^{- m} = \frac{e^{- m} m^{r}}{r!} \end{array}$