query: Is the product of two factorial always smaller than the factorial of the sum?
positive_passages.docid: 153763
positive_passages.text: Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
negative_passages.docid: 41789
negative_passages.text: Fermat's theorem on sums of two squares
negative_passages.docid: 17638
negative_passages.text: Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
negative_passages.docid: 144439
negative_passages.text: For a $C^1$ function, the difference $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |$ is small when $|d-c|$ is small
negative_passages.docid: 90447
negative_passages.text: Dividing factorials is always integer
negative_passages.docid: 147111
negative_passages.text: Where is the Proof for the Sum-to-Product rule derrived from?
negative_passages.docid: 18023
negative_passages.text: The direct sum $\oplus$ versus the cartesian product $\times$
negative_passages.docid: 57986
negative_passages.text: Sum of products of combinatorials
negative_passages.docid: 153239
negative_passages.text: Why computing factorial of realtively small numbers (34+) returns 0
negative_passages.docid: 87479
negative_passages.text: Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
negative_passages.docid: 69228
negative_passages.text: Factorial of 0 - a convenience?
negative_passages.docid: 103572
negative_passages.text: Does $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}^3=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}a_{mn}^3$ occur here?
negative_passages.docid: 40784
negative_passages.text: $\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
negative_passages.docid: 71500
negative_passages.text: Is the product of generators equal to the generator of the product?
negative_passages.docid: 20618
negative_passages.text: Expressing a factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?
negative_passages.docid: 21986
negative_passages.text: What is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$?
negative_passages.docid: 68080
negative_passages.text: Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$