26.3

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
26.3.E1	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\genfrac{(}{)}{0.0pt}{}{m}{m-n}}}$ \binom{m}{n} = \binom{m}{m-n}	${\displaystyle{\displaystyle m\geq n}}$	binomial(m,n) = binomial(m,m - n)	Binomial[m,n] == Binomial[m,m - n]	Failure	Successful	Successful [Tested: 6]	Successful [Tested: 6]
26.3.E1	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{m-n}=\frac{m!}{(m-n)!\,n!}}}$ \binom{m}{m-n} = \frac{m!}{(m-n)!\,n!}	${\displaystyle{\displaystyle m\geq n}}$	binomial(m,m - n) = (factorial(m))/(factorial(m - n)*factorial(n))	Binomial[m,m - n] == Divide[(m)!,(m - n)!*(n)!]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 6]
26.3.E2	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=0}}$ \binom{m}{n} = 0	${\displaystyle{\displaystyle n>m}}$	binomial(m,n) = 0	Binomial[m,n] == 0	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
26.3.E3	${\displaystyle{\displaystyle\sum_{n=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{n}x^{n}=(1+x)^{m}}}$ \sum_{n=0}^{m}\binom{m}{n}x^{n} = (1+x)^{m}		sum(binomial(m,n)*(x)^(n), n = 0..m) = (1 + x)^(m)	Sum[Binomial[m,n]*(x)^(n), {n, 0, m}, GenerateConditions->None] == (1 + x)^(m)	Successful	Successful	-	Successful [Tested: 0]
26.3.E4	${\displaystyle{\displaystyle\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{m+n}{m}x^{m}=\frac{1}{(1-x)^{n+1}}}}$ \sum_{m=0}^{\infty}\binom{m+n}{m}x^{m} = \frac{1}{(1-x)^{n+1}}	${\displaystyle{\displaystyle|x|<1}}$	sum(binomial(m + n,m)*(x)^(m), m = 0..infinity) = (1)/((1 - x)^(n + 1))	Sum[Binomial[m + n,m]*(x)^(m), {m, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - x)^(n + 1)]	Successful	Successful	-	Successful [Tested: 3]
26.3.E5	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\genfrac{(}{)}{0.0pt}{}{m-1}{n}+\genfrac{(}{)}{0.0pt}{}{m-1}{n-1}}}$ \binom{m}{n} = \binom{m-1}{n}+\binom{m-1}{n-1}	${\displaystyle{\displaystyle m\geq n,n\geq 1}}$	binomial(m,n) = binomial(m - 1,n)+binomial(m - 1,n - 1)	Binomial[m,n] == Binomial[m - 1,n]+Binomial[m - 1,n - 1]	Successful	Successful	-	Successful [Tested: 6]
26.3.E6	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\frac{m}{n}\genfrac{(}{)}{0.0pt}{}{m-1}{n-1}}}$ \binom{m}{n} = \frac{m}{n}\binom{m-1}{n-1}	${\displaystyle{\displaystyle m\geq n,n\geq 1}}$	binomial(m,n) = (m)/(n)*binomial(m - 1,n - 1)	Binomial[m,n] == Divide[m,n]*Binomial[m - 1,n - 1]	Successful	Successful	-	Successful [Tested: 6]
26.3.E6	${\displaystyle{\displaystyle\frac{m}{n}\genfrac{(}{)}{0.0pt}{}{m-1}{n-1}=\frac{m-n+1}{n}\genfrac{(}{)}{0.0pt}{}{m}{n-1}}}$ \frac{m}{n}\binom{m-1}{n-1} = \frac{m-n+1}{n}\binom{m}{n-1}	${\displaystyle{\displaystyle m\geq n,n\geq 1}}$	(m)/(n)*binomial(m - 1,n - 1) = (m - n + 1)/(n)*binomial(m,n - 1)	Divide[m,n]*Binomial[m - 1,n - 1] == Divide[m - n + 1,n]*Binomial[m,n - 1]	Successful	Successful	-	Successful [Tested: 6]
26.3.E7	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m+1}{n+1}=\sum_{k=n}^{m}\genfrac{(}{)}{0.0pt}{}{k}{n}}}$ \binom{m+1}{n+1} = \sum_{k=n}^{m}\binom{k}{n}	${\displaystyle{\displaystyle m\geq n,n\geq 0}}$	binomial(m + 1,n + 1) = sum(binomial(k,n), k = n..m)	Binomial[m + 1,n + 1] == Sum[Binomial[k,n], {k, n, m}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 6]
26.3.E8	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{m-n-1+k}{k}}}$ \binom{m}{n} = \sum_{k=0}^{n}\binom{m-n-1+k}{k}	${\displaystyle{\displaystyle m\geq n,n\geq 0}}$	binomial(m,n) = sum(binomial(m - n - 1 + k,k), k = 0..n)	Binomial[m,n] == Sum[Binomial[m - n - 1 + k,k], {k, 0, n}, GenerateConditions->None]	Successful	Successful	-	Failed [3 / 6] Result: Indeterminate Test Values: {Rule[m, 1], Rule[n, 1]} Result: Indeterminate Test Values: {Rule[m, 2], Rule[n, 2]} ... skip entries to safe data
26.3.E9	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{0}=\genfrac{(}{)}{0.0pt}{}{n}{n}}}$ \binom{n}{0} = \binom{n}{n}		binomial(n,0) = binomial(n,n)	Binomial[n,0] == Binomial[n,n]	Successful	Successful	-	Successful [Tested: 3]
26.3.E9	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{n}=1}}$ \binom{n}{n} = 1		binomial(n,n) = 1	Binomial[n,n] == 1	Successful	Successful	-	Successful [Tested: 3]
26.3.E10	${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\sum_{k=0}^{n}(-1)^{n-k}\genfrac{(}{)}{0.0pt}{}{m+1}{k}}}$ \binom{m}{n} = \sum_{k=0}^{n}(-1)^{n-k}\binom{m+1}{k}	${\displaystyle{\displaystyle m\geq n,n\geq 0}}$	binomial(m,n) = sum((- 1)^(n - k)*binomial(m + 1,k), k = 0..n)	Binomial[m,n] == Sum[(- 1)^(n - k)*Binomial[m + 1,k], {k, 0, n}, GenerateConditions->None]	Successful	Failure	-	Successful [Tested: 6]