Prove that the sum of the binomial coefficients from 0 to n equals 2^n using a combinatorial argument about bit strings.

Vault Of Euclid

Sum of Binomial Coefficients via Bit Strings

Counting warmup

Let \( n \) be a positive integer. Explain why \[ \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \dots + \binom{n}{n} = 2^n. \]

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What you need first

The definition of binomial coefficients (combinations).
The multiplication principle of counting.
The concept of a combinatorial proof.

What this problem tests

Tests the student's ability to construct a combinatorial proof by identifying a set of objects and counting it in two distinct ways.

This is a fundamental identity in combinatorics, often introduced early in discrete mathematics courses to illustrate the power of combinatorial proofs over algebraic manipulation.

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