[QUOTE=R.D. Silverman;219777]He had stated earlier that he was familiar with the binomial theorem........[/QUOTE]

Apparently not familiar enough. :grin:

[QUOTE=jyb;219778]Apparently not familiar enough. :grin:[/QUOTE]

I mean [b]no[/b] insult to Tomer in what I am going to say,
but he really thinks that he knows a lot of things that he really
doesn't know. It is surely not his fault; I'm sure that the training
he received has been inadequate to really do proofs.

He seems to have very poor training in the use of definitions, use
of variables, substitution of variables, and his derivations just lack
rigor. He would benefit greatly from a disciplined proof-based course
in Eulclidean Geometry.

[quote=R.D. Silverman;219781]I mean [B]no[/B] insult to Tomer in what I am going to say,
but he really thinks that he knows a lot of things that he really
doesn't know. It is surely not his fault; I'm sure that the training
he received has been inadequate to really do proofs.

He seems to have very poor training in the use of definitions, use
of variables, substitution of variables, and his derivations just lack
rigor. He would benefit greatly from a disciplined proof-based course
in Eulclidean Geometry.[/quote]

"but he really thinks that he knows a lot of things that he really
doesn't know." False.
I didn't say I know something about the binom (I asked about it in one of the posts, becuase I know it's idea and theme, but I lack skills on the area).

And now, after reading Wolfram's page on the binomial coefficients, I may say it is n_C_k=n!/(k!(n-k)!).

[QUOTE=blob100;219787]n_C_k=n!/(k!(n-k)!).[/QUOTE]

Also meaning no offense, are you able to formulate a proof?

At least to me, you have yet to demonstrate that you have the skills required.

[QUOTE=blob100;219787]And now, after reading Wolfram's page on the binomial coefficients, I may say it is n_C_k=n!/(k!(n-k)!).[/QUOTE]

Okay, you may [I]say[/I] it, but do you [I]understand[/I] it? In what circumstances might it be useful? Can you see how it applies to the polynomial coefficient problem? This is what I meant when I said that you needed to "think about this information".

[quote=Wacky;219788]Also meaning no offense, are you able to formulate a proof?

At least to me, you have yet to demonstrate that you have the skills required.[/quote]

I'll try.
Is induction the right approach to do so?

[QUOTE=blob100;219790]I'll try.
Is induction the right approach to do so?[/QUOTE]

"All roads lead to Rome" -- As with many such problems, there are multiple ways to formulate a proof. Any of them would be acceptable.

A proof by induction is certainly a reasonable approach.

[quote=Wacky;219791]"All roads lead to Rome" -- As with many such problems, there are multiple ways to formulate a proof. Any of them would be acceptable.

A proof by induction is certainly a reasonable approach.[/quote]
OK.

[QUOTE=R.D. Silverman;219781]He seems to have very poor training in the use of definitions, use
of variables, substitution of variables, and his derivations just lack
rigor. He would benefit greatly from a disciplined proof-based course
in Eulclidean Geometry.[/QUOTE]

I would say that Tomer likes math and has discovered a number of interesting things about it which make him eager to learn more. Being bright and eager is a good start, but it will only get one so far. In order to really understand math, one also needs to learn rigor, and that requires discipline (as you've said many times). The key is to impart that sense of rigor and discipline without destroying the fun which makes one eager in the first place.

He is untrained and lacks rigor. Well that's not a fatal flaw. I think he just hasn't been exposed to that way of thinking much. All of your complaints about his definitions, variables, derivations, etc. are really just symptoms of this lack of exposure. Learning to do proofs is a bit like learning a foreign language; to some degree there's just a style to which one must become accustomed, and being immersed in that is the best way to learn it.

You're probably right about a course in geometry, partly because he just needs to spend some time learning to do rigorous proofs, and partly because geometry is a good vehicle for maintaining that sense of fun: who doesn't like drawing pictures?

Of course there's really no way to give him geometry problems in this forum, but there are other kinds of problems we could give him that are very simple and would help him practice proofs.

Proof by induction (binomial theorem):
We see define [sub]n[/sub]C[sub]k[/sub] as the number of combinations of k units taken from a set of n units.

Proposition:
[sub]n[/sub]C[sub]k[/sub]=(n!)/(P!k!)
P=n-k.

We will do so by induction on n.
Step 1:
Showing [sub]0[/sub]C[sub]k[/sub] agree the terms (for n=0, the proposition is true).
[sub]0[/sub]C[sub]k[/sub]=1.
(0!)/(P!k!)=1/(1*1)=1.

Step 2:
Assuming that the formula is true for a given n and all 0≤ k≤n.

Step 3:
Showing that for n+1 the proposition is true if it is for n.
We assumed:
[sub]n[/sub]C[sub]k[/sub]=(n!)/(P!k!),
And we try to prove:
[sub]n+1[/sub]C[sub]k[/sub]=((n+1)!)/(P!k!).
We see: k=0, ((n+1)!)/((n+1)!0!)=1.
And: k=n+1, ((n+1)!)/(0!(n+1)!)=1
And for 0≤ k≤n, [sub]n[/sub]C[sub]k[/sub]+[sub]n[/sub]C[sub]k-1[/sub].
So, we try porving [sub]n[/sub]C[sub]k[/sub]+[sub]n[/sub]C[sub]k-1[/sub] is equivallent to (n+1)!/((n+1-k)!k!).
n!/(n-k)!k!+n!/(n-k+1)!(k-1)!=n!(n-k+1)/(n-k+1)!k!+n!k/(n-k+1)!k!=
n!(n+1)/(n-k+1)!k!.
Now, we would easily propose the next equation:
n!(n+1)/(n-k+1)!k!=(n+1)!/((n+1-k)!k!),
Which is what we tried to prove.

[quote=jyb;219793]I would say that Tomer likes math and has discovered a number of interesting things about it which make him eager to learn more. Being bright and eager is a good start, but it will only get one so far. In order to really understand math, one also needs to learn rigor, and that requires discipline (as you've said many times). The key is to impart that sense of rigor and discipline without destroying the fun which makes one eager in the first place.

He is untrained and lacks rigor. Well that's not a fatal flaw. I think he just hasn't been exposed to that way of thinking much. All of your complaints about his definitions, variables, derivations, etc. are really just symptoms of this lack of exposure. Learning to do proofs is a bit like learning a foreign language; to some degree there's just a style to which one must become accustomed, and being immersed in that is the best way to learn it.

You're probably right about a course in geometry, partly because he just needs to spend some time learning to do rigorous proofs, and partly because geometry is a good vehicle for maintaining that sense of fun: who doesn't like drawing pictures?

Of course there's really no way to give him geometry problems in this forum, but there are other kinds of problems we could give him that are very simple and would help him practice proofs.[/quote]

I think Discrete mathematics is as Euclidean Geometry a good course to learn proofs.
Do you agree with me?