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Thread: Binomial coefficients

  1. #1 Binomial coefficients 
    Forum Masters Degree thyristor's Avatar
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    Ever thought of that the binomial koefficients equal 11<sup>n</sup>?


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  3. #2  
    Forum Professor serpicojr's Avatar
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    Eh what?


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  4. #3  
    Forum Ph.D.
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    The sum of binomial coefficients is equal to 2<sup>n</sup>. For example, <sup>5</sup>C<sub>0</sub> + <sup>5</sup>C<sub>1</sub> + <sup>5</sup>C<sub>2</sub> + <sup>5</sup>C<sub>3</sub> + <sup>5</sup>C<sub>4</sub> + <sup>5</sup>C<sub>5</sub> = 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2<sup>5</sup>.

    Proof: Take x = 1 in the binomial expansion of (1+x)<sup>n</sup>.
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  5. #4  
    Forum Professor serpicojr's Avatar
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    Yeah, I thought that's perhaps what he meant.
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  6. #5  
    Forum Radioactive Isotope MagiMaster's Avatar
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    I think he meant that the digits of 11^n are binomial coefficients.
    1
    11
    121
    1331
    14641
    161051 <- It breaks here though

    101^n lasts a while longer. In general, any 10^k+1 works for a while and the reason is pretty simple. (a + b)^n = sum over i of (n choose i)*(a^i)*(b^(n-i)), so for a=10, b=1 you're just left with powers of 10 times the binomial coefficients.
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  7. #6  
    Forum Masters Degree thyristor's Avatar
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    You are right about what I meant Magi Master but the only reason that the binomial coefficients aren't palindroms all the time even if they equal 11<sup>n</sup> is that ten symbols, as we use, aren't enough.
    There will be overflow.
    If you want to you can write a computer program with for example 15 symbols and there you'll see that 11<sup>5</sup> is a palindrom.
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  8. #7  
    Forum Radioactive Isotope MagiMaster's Avatar
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    Yes, in a higher base, 11<sup>n</sup> will stay a palindrome for longer, but never indefinitely (101<sup>n</sup> is about the same as 11<sup>n</sup> in base-100). I suppose you could define something like a base-infinity (based on tuples) that would stay a palindrome but I don't think there's much point in that (correct me if I'm wrong).
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  9. #8  
    Forum Masters Degree thyristor's Avatar
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    The only point is that palindromes are beautiful.
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