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![[tex]{a \choose b} = \frac{a!}{b!(a - b)!}[/tex]](images/latex/9adff991243b9cada18f6d6914d66debd16947c9_0.png "[tex]{a \choose b} = \frac{a!}{b!(a - b)!}[/tex]")
entonces con a = 2k + 1 y b = 2j + 1:
![[tex]{2k+1 \choose 2j + 1} = \frac{(2k+1)!}{(2j+1)!(2k+1-2j-1)!} = \frac{(2k+1)!}{(2j+1)![2(k-j)]!}[/tex]](images/latex/fdf1eb71e59fd0c60bbbb2c49ebe146d2a606ded_0.png "[tex]{2k+1 \choose 2j + 1} = \frac{(2k+1)!}{(2j+1)!(2k+1-2j-1)!} = \frac{(2k+1)!}{(2j+1)![2(k-j)]!}[/tex]")
![[tex]{2k+1 \choose 2(k-j)} = \frac{(2k+1)!}{[2(k-j)]![2k+1-2(k-j)]!} = \frac{(2k+1)!}{[2(k-j)]!(2j+1)!}[/tex]](images/latex/1d339a475f148e63f5a6a7e8302eb9ae8d6c22d3_0.png "[tex]{2k+1 \choose 2(k-j)} = \frac{(2k+1)!}{[2(k-j)]![2k+1-2(k-j)]!} = \frac{(2k+1)!}{[2(k-j)]!(2j+1)!}[/tex]")
![[tex]\sum_{j=0}^k {2k+1 \choose 2j+1} =\frac{1}{2}\sum_{j=0}^k {2k+1 \choose 2(k-j)} + \frac{1}{2}\sum_{j=0}^k {2k+1 \choose 2j+1}[/tex]](images/latex/6cf94ac808eb5d8d267a6f6783ee5f46e707e653_0.png "[tex]\sum_{j=0}^k {2k+1 \choose 2j+1} =\frac{1}{2}\sum_{j=0}^k {2k+1 \choose 2(k-j)} + \frac{1}{2}\sum_{j=0}^k {2k+1 \choose 2j+1}[/tex]")
![[tex]\sum_{r=0}^k {2k+1 \choose 2r} = {2k+1 \choose 0} + {2k+1 \choose 2} + {2k+1 \choose 4} + \cdots + {2k+1 \choose 2k}[/tex]](images/latex/464f8f1e684bed1d3322fefe270d82b3f3aefb12_0.png "[tex]\sum_{r=0}^k {2k+1 \choose 2r} = {2k+1 \choose 0} + {2k+1 \choose 2} + {2k+1 \choose 4} + \cdots + {2k+1 \choose 2k}[/tex]")
![[tex]{2k+1 \choose q}[/tex]](images/latex/01ffacec848391d2a5fa34e174a4e86c059cac79_0.png "[tex]{2k+1 \choose q}[/tex]")
con ![[tex]\sum_{j=0}^k {2k+1 \choose 2j+1} = {2k+1 \choose 1} + {2k+1 \choose 3} + {2k+1 \choose 5} + \cdots + {2k+1 \choose 2k+1}[/tex]](images/latex/2c66121e09f2f030ae995c90cde8bfda75b2e685_0.png "[tex]\sum_{j=0}^k {2k+1 \choose 2j+1} = {2k+1 \choose 1} + {2k+1 \choose 3} + {2k+1 \choose 5} + \cdots + {2k+1 \choose 2k+1}[/tex]")
![[tex]{2k+1 \choose 0}[/tex]](images/latex/42b4e1e02b200638dba6cb4afee419df5e779cf8_0.png "[tex]{2k+1 \choose 0}[/tex]")
hasta ![[tex]{2k+1 \choose 2k+1}[/tex]](images/latex/684815a51f7d5ebea9c90ed9da5e5da4cf1f6403_0.png "[tex]{2k+1 \choose 2k+1}[/tex]")
:
![[tex]\frac{1}{2}\sum_{j=0}^k {2k+1 \choose 2j+1} + \frac{1}{2}\sum_{r=0}^k {2k+1 \choose 2r} =\frac{1}{2}\sum_{j=0}^{2k+1}{2k+1 \choose j}[/tex]](images/latex/2ad84f181da922cefeb6e9f5fd9c99d0e8e4df71_0.png "[tex]\frac{1}{2}\sum_{j=0}^k {2k+1 \choose 2j+1} + \frac{1}{2}\sum_{r=0}^k {2k+1 \choose 2r} =\frac{1}{2}\sum_{j=0}^{2k+1}{2k+1 \choose j}[/tex]")
^n = \sum_{j=0}^n {n \choose j} x^{n-j}y^j[/tex]](images/latex/c0711f321a7c456913b7437f14e7ae57ac5f3611_0.png "[tex](x + y)^n = \sum_{j=0}^n {n \choose j} x^{n-j}y^j[/tex]")
![[tex]\sum_{j=0}^{2k+1} {2k+1 \choose j} = 2^{2k+1}[/tex]](images/latex/4ad6f296d776842a5b0d49e8a15c9f7e34811fc3_0.png "[tex]\sum_{j=0}^{2k+1} {2k+1 \choose j} = 2^{2k+1}[/tex]")
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