Math Problem Statement
Show that $\sum_{k=0}^n \binom{k}{i} = \binom{n+1}{i+1}$ for all integers $n\geq 0$ and $i\geq 0$. Again, use a combinatorial argument, where you show that both sides count the same thing.
Solution
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Binomial Coefficients
Summation
Formulas
\(\sum_{k=0}^n \binom{k}{i} = \binom{n+1}{i+1}\)
Pascal's Identity: \(\binom{n+1}{i+1} = \binom{n}{i} + \binom{n}{i+1}\)
Theorems
Pascal's Identity
Binomial Theorem
Suitable Grade Level
Grades 11-12 or Undergraduate
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