More Binomial... (2 Viewers)

[micuzzo]

can someone please show me how to prove nCr = nCr+1 = n+1Cr+1 without induction... and can you please have full working out

most apriciated

 

[Trebla]

Ok first of all: ⁿC_r = ⁿC_r+1 IS FALSE!!! Think about it, there's no way ⁶C₁ = ⁶C₂ for example. Similarly for ^{n + 1}C_r+1.

I think you meant: ⁿC_r + ⁿC_{r + 1} = ^{n + 1}C_{r + 1}
LHS = ⁿC_r + ⁿC_r+1
= n! / [r!(n - r)!] + n! / {(r + 1)!(n - [r + 1])!}
= n! / [r!(n - r)!] + n! / {(r + 1)!(n - r - 1)!}
= n! / [r!(n - r - 1)!] { 1 / (n - r) + 1 / (r + 1) }
= n! / [r!(n - r - 1)!] x {r + 1 + n - r} / (n - r)(r + 1)
= n! / [r!(n - r - 1)!] x {n + 1} / (n - r)(r + 1)
= (n + 1)! / [(r + 1)!(n - r)!]
= (n + 1)! / [(r + 1)!(n + 1 - 1 - r)!]
= (n + 1)! / [(r + 1)!(n + 1 - {r + 1})!]
= ^{n + 1}C_{r + 1}
= RHS

 

[bored of sc]

Wow you are smart Trebla. Give me your brain.

 

[micuzzo]

Ok first of all: ⁿC_r = ⁿC_r+1 IS FALSE!!! Think about it, there's no way ⁶C₁ = ⁶C₂ for example. Similarly for ^{n + 1}C_r+1.

I think you meant: ⁿC_r + ⁿC_{r + 1} = ^{n + 1}C_{r + 1}
LHS = ⁿC_r + ⁿC_r+1
= n! / [r!(n - r)!] + n! / {(r + 1)!(n - [r + 1])!}
= n! / [r!(n - r)!] + n! / {(r + 1)!(n - r - 1)!}
= n! / [r!(n - r - 1)!] { 1 / (n - r) + 1 / (r + 1) }
= n! / [r!(n - r - 1)!] x {r + 1 + n - r} / (n - r)(r + 1)
= n! / [r!(n - r - 1)!] x {n + 1} / (n - r)(r + 1)
= (n + 1)! / [(r + 1)!(n - r)!]
= (n + 1)! / [(r + 1)!(n + 1 - 1 - r)!]
= (n + 1)! / [(r + 1)!(n + 1 - {r + 1})!]
= ^{n + 1}C_{r + 1}
= RHS

umm i forgot about this post lol... one question... i dont understand the third line and also, how/ why does (n-r-1)! x (n-r) = (n-r)! <--- ive never really understood that.

 

[lyounamu]

umm i forgot about this post lol... one question... i dont understand the third line and also, how/ why does (n-r-1)! x (n-r) = (n-r)1 <--- ive never really understood that.

4 x 3! = 4!

yes?

Then y x (y-1)! = y! ?

then (n-r) x (n-r-1)! = (n-r)!

 

[micuzzo]

that was fast

4 x 3! = 4!

yes?

Then y x (y-1)! = y! ?

then (n-r) x (n-r-1)! = (n-r)!

hmmm... i think you may be onto somthing here... lol

thanks.....

as if i didnt think of that *whispers to self*

 

[lyounamu]

that was fast




hmmm... i think you may be onto somthing here... lol

thanks.....

as if i didnt think of that *whispers to self*

okay, i guess this explanation would be slightly better:

4! x 5 = (1 x 2 x 3 x 4) x 5 a
= 1 x 2 x 3 x 4 x 5 = 5!

so

(x-1)! multiplied by x = x!

 

[micuzzo]

nah its all good... i understood it the first time... oh congrats on comming first in state dude