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IloveManUtd
 Post subject: Binomial expansions
PostPosted: Mon, 13 Sep 2010 02:46:50 UTC 
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If the coefficients of x^k and x^(k+1) in the expansion of (2+3x)^19 are equal, find k.
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robbwrr
 Post subject: Re: Binomial expansions
PostPosted: Mon, 13 Sep 2010 03:07:18 UTC 
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IloveManUtd wrote:
If the coefficients of x^k and x^(k+1) in the expansion of (2+3x)^19 are equal, find k.


$ \frac{19! 2^{(19-k)} 3^k}{k! (19-k)!}=\frac{19! 2^{(19-k-1)} 3^{(k+1)}}{(k+1)! (19-k-1)!}

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IloveManUtd
 Post subject: Re: Binomial expansions
PostPosted: Mon, 13 Sep 2010 03:10:46 UTC 
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robbwrr wrote:

$ \frac{19! 2^{(19-k)} 3^k}{k! (19-k)!}=\frac{19! 2^{(19-k-1)} 3^{(k+1)}}{(k+1)! (19-k-1)!}


Can you please explain what how do I make k! and (19-k)! more meaningful? Thx!
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outermeasure
 Post subject: Re: Binomial expansions
PostPosted: Mon, 13 Sep 2010 14:49:07 UTC 
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IloveManUtd wrote:
robbwrr wrote:

$ \frac{19! 2^{(19-k)} 3^k}{k! (19-k)!}=\frac{19! 2^{(19-k-1)} 3^{(k+1)}}{(k+1)! (19-k-1)!}


Can you please explain what how do I make k! and (19-k)! more meaningful? Thx!


Cancel common factors.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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