First slide

Binomial theorem for positive integral Index

Question

If $C_{r} =^{n} C_{r},$ then the value of $(1 + \frac{C_{1}}{C_{0}}) (1 + \frac{C_{2}}{C_{1}}) \dots (1 + \frac{C_{n}}{C_{n - 1}})$

Moderate

A

$\frac{(n + 1)!}{2^{n}}$
B

$\frac{(n + 1)^{n}}{n!}$
C

$\frac{2^{n - 1}}{n!}$
D

$\frac{2^{n} - 1}{n!}$

Solution

From the previous question, we have
$(1 + \frac{C_{r}}{C_{r - 1}}) = \frac{n + 1}{r}$
$∴ (1 + \frac{C_{1}}{C_{0}}) (1 + \frac{C_{2}}{C_{1}}) \dots (1 + \frac{C_{n}}{C_{n - 1}}) = \frac{(n + 1)^{n}}{n!}$

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