Binomial Coefficient

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Binomial Coefficient

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#1 h May 19, 2021, 8:43 AM • 2 Y

Y by samrocksnature, centslordm

How many integer number $1\le k\le 2010$ such that $4\mid\binom{2010}{k}$ ?

This post has been edited 1 time. Last edited by Wildabandon, May 19, 2021, 8:43 AM

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nibaba

#2 h May 19, 2021, 9:08 AM • 2 Y

Y by samrocksnature, centslordm

Kummer theory can easily solve this problem.

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math31415926535

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math31415926535

#3 h May 19, 2021, 2:55 PM • 2 Y

Y by samrocksnature, centslordm

what is that?

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8219 posts

#4 h May 19, 2021, 3:08 PM • 3 Y

Y by samrocksnature, centslordm, Mango247

nibaba wrote:

Kummer theory can easily solve this problem.

I don't think you need Kummer's theorem. Lucas's theorem should be sufficient.

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#5 h May 19, 2021, 3:14 PM • 2 Y

Y by samrocksnature, centslordm

I believe only odd values work except for $1, 2009$ , so $3,5,\ldots 2007 \implies \boxed{1003}$ .

This post has been edited 1 time. Last edited by OlympusHero, May 19, 2021, 3:15 PM

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#7 h May 19, 2021, 3:57 PM • 3 Y

Y by hurdler, samrocksnature, centslordm

@above $\binom{2010}{9}$ and $\binom{2010}{12}$ are counter examples to your claim.
solution

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#8 h May 20, 2021, 2:33 AM • 5 Y

Y by samrocksnature, natmath, pog, Wildabandon, centslordm

I think the answer should be $2010-(2^8-1)-(3\times 2^7)$ .

The p-adic valuation for Pascal's triangle row

n=0
for k in range(1,2010+1):
 p=0
 for m in range(1,20):
  p=p+int(2010/2**m)-int(k/2**m)-int((2010-k)/2**m)
 if p>=2:
  n=n+1
print(n)

n=0
for k in range(1,2010+1):
 p=0
 for m in range(1,20):
  p=p+int(2010/2**m)-int(k/2**m)-int((2010-k)/2**m)
 if p>=2:
  n=n+1
print(n)

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#9 h May 20, 2021, 5:26 AM • 1 Y

Y by centslordm

natmath wrote:

@above $\binom{2010}{9}$ and $\binom{2010}{12}$ are counter examples to your claim.
solution

Oops, I forgot that Lucas' theorem only applies to primes. Yeah, we have to use Kummer's theorem then.

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#10 h May 20, 2021, 6:07 AM • 2 Y

Y by centslordm, Mango247

I try use Kummer's Theorem. I think is still bashy to got $S(k) + S(2020-k) \ge 9$ where $S(n)$ to be the sum of the base- $2$ digits of $n$ .

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#11 h May 20, 2021, 11:11 AM • 2 Y

Y by centslordm, Mango247

Wait, I see someone has done more than me in 2008.

NUMBER OF BINOMIAL COEFFICIENTS DIVISIBLE BY A FIXED
POWER OF A PRIME

His special cases for $\theta_0(n,p),~\theta_1(n,p)$ are exactly what I wrote.

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#12 h May 24, 2021, 1:46 AM • 1 Y

Y by centslordm

@above Nice blog. Thank you!

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