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Proposition: Simple Binomial Identities

The following identities hold for binomial coefficients:

1. $$k\binom nk=n\binom {n-1}{k-1}\quad\quad\text{for }k\ge 1.$$
2. $$k^2\binom nk=k\binom nk+n(n-1)\binom {n-2}{k-2}\quad\quad\text{for }k\ge 2.$$
3. $$k^2\binom nk=k\binom nk+n(n-1)\binom {n-2}{k-2}\quad\quad\text{for }k\ge 2.$$

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References

Bibliography

1. Bosch, Karl: "Elementare Einführung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition