Combinatorial identity's algebraic proof without induction.
8476
Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
25004
Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
18301
Proof by Induction involving Inequality and Factorials as denominators
107021
Finding a combinatorial argument for an interesting identity: $\sum_k \binom nk \binom{m+k}n = \sum_i \binom ni \binom mi 2^i$
117775
Combinatorial proof of a Stirling number identity.
90904
Combinatorial proof of an identity with squared binomial coefficients
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
60712
Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares
15100
Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction
32081
The dual version of trichotomy of cardinality without AC
96903
How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $?
158728
Using induction for an easy proof for formal languages
11335
Proving Binomial Identity without calculus
4610
Proving an identity with a combinatorial proof: $\binom{n}{k}\binom{k}{r}=\binom{n}{r}\binom{n-r}{k-r}$
17580
Combinatorics That Looks Similar to Vandermonde's Identity
71564
Combinatorial proof of identity involving central binomial coefficients