Binomial Theorem Applications

Identity Proof: Start with the standard expansion:
- $(1 + x)^{n} = (0 n) + (1 n) x + (2 n) x^{2} + \dots + (n n) x^{n}$
Substitution: Let $x = 1$ .
- $(1 + 1)^{n} = (0 n) + (1 n) (1) + (2 n) (1)^{2} + \dots + (n n) (1)^{n}$
Result: $2^{n} = \sum_{r = 0}^{n} (r n)$ . The sum of all binomial coefficients for a power $n$ is $2^{n}$ .

This easy difficulty mathematics question is from the chapter binomial theorem and its simple applications, covering the topic of binomial theorem applications. It appeared in the 2025 exam.

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Binomial Theorem Applications

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