The Gateway to Computer Science Excellence
+1 vote
128 views

Let $(1+x)^n = C_0+C_1x+C_2x^2+ \dots + C_nx^n$, $n$ being a positive integer. The value of

$$\bigg( 1+\dfrac{C_0}{C_1} \bigg) \bigg( 1+\dfrac{C_1}{C_2} \bigg) \cdots \bigg( 1+\dfrac{C_{n-1}}{C_n} \bigg)$$ is

  1. $\bigg( \frac{n+1}{n+2} \bigg) ^n$
  2. $ \frac{n^n}{n!} $
  3. $\bigg( \frac{n}{n+1} \bigg) ^n$
  4. $ \frac{(n+1)^n}{n!} $
in Combinatory by Veteran (424k points)
recategorized by | 128 views

2 Answers

+5 votes

(D) $\frac{(n+1)^n}{n!}$

Here $C_p=$$\binom{n}{p}$

Then,$\frac{C_p}{C_{p+1}}=\frac{p+1}{n-p}$

Then,$1+\frac{C_p}{C_{p+1}}=\frac{n+1}{n-p}$

For p=0 $=>$ $1+\frac{C_0}{C_{1}}=\frac{n+1}{n}$

For p=1 $=>$ $1+\frac{C_1}{C_{2}}=\frac{n+1}{n-1}$

and so on..

Thus if we multiply all the terms we get $\frac{(n+1)^n}{n!}$

by Active (1.3k points)
+1
Put n=1 then(1+1)=2

Check in answer Option D satisfied
0 votes
If we take the value of N==3, then from the condition we get the value of expression as 32/3.

 

Now checking the options when we put N==3 we must get the same value and this is satisfied only by option D put n==3 we get 4^3/Fact(3) which is equal to 32/3 hence the correct option is Option D.
by (91 points)

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,647 questions
56,492 answers
195,440 comments
100,712 users