Pascal's Triangle

Question

In Pascal's triangle, the sum of elements in the nth row is

RankGuru · Accepted Answer

1. **Observation**: Pascal's triangle represents binomial coefficients.
   - Row 0: $1$ (Sum $= 1 = 2^0$)
   - Row 1: $1, 1$ (Sum $= 2 = 2^1$)
   - Row 2: $1, 2, 1$ (Sum $= 4 = 2^2$)
   - Row 3: $1, 3, 3, 1$ (Sum $= 8 = 2^3$)
2. **Math Property**: The $n^{th}$ row consists of $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$.
3. **Result**: As proven in the identity $\sum \binom{n}{r} = 2^n$, the sum is **$2^n$**.

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Pascal's Triangle

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