Proof of Binomial Distribution by Induction

the question ive been asked is..

Show C(n,r), defined by

C(n, r) = n! / r!(n-r)! , n ∈ N , r = 0,1... n

always lies in N using a proof by induction.

i need help, i have a rough idea about proof by induction but i dont no how to apply it to this question. can anybody help me?

Reply 1

nota bene

15

You probably want to state that r<=n somewhere as well.

Then start by showing it holds for n=1 (=>r=1 or r=0 and r=0 is trivial/not very interesting)

Assume it holds for n. Then try finding a relationship between the n+1 case and n case.

Please post back your attempt and we can give more specific help.

Reply 2

sarah_vickers

OP

ive sed n=1 and r=0...

1! / 0!(1-0)! which equals 1.

when i put it is terms of k: n=k and then n=k+1 i get confused on how to work it out. in k+1 do i make r=0 again, or do i move onto the next number, which is r=0? i am unsure oon the whole question to be honest lol

Reply 3

sarah_vickers

OP

ive put in n=1 and r=0 to get 1. i also put in n=1, r=1 to get 1 also. i then wrote it in terms of k where n=k. then i put n=k+1. wen i put r=1 into the k+1, i get:

(k+1)! / k! which proves that it always lies in N.

is this correct because im not sure about it, nobody else in my group no's if it is correct either?

Reply 4

DFranklin

18

I would say the simplest thing is to work from the identity

\binom{n+1}{k} = \binom{n}{k}+\binom{n}{k-1}

(and use induction on n).

Reply 5

sarah_vickers

OP

DFranklin

I would say the simplest thing is to work from the identity

\binom{n+1}{k} = \binom{n}{k}+\binom{n}{k-1}

(and use induction on n).

im not sure i understand... can u be abit more specific please? thanks :smile:

Reply 6

DFranklin

18

Well, if you're using induction on n, you know the two terms on the RHS are integers (by the induction hypothesis). So, what can we say about the LHS?

Reply 7

sarah_vickers

OP

DFranklin

Well, if you're using induction on n, you know the two terms on the RHS are integers (by the induction hypothesis). So, what can we say about the LHS?

i really dont understand... y r the two terms intergers? wat can i say about the LHS?

Reply 8

DFranklin

18

Please don't use text speak, it is against the forum rules.

As far as the question goes - I think you'd be best speaking to your teacher/lecturer. It looks like you are very confused, and they need to know that.

Reply 9

sarah_vickers

OP

DFranklin

Please don't use text speak, it is against the forum rules.

As far as the question goes - I think you'd be best speaking to your teacher/lecturer. It looks like you are very confused, and they need to know that.

its not that, its just my teacher never taught me that equation u shows me in the previous post.

Proof of Binomial Distribution by Induction

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