Let the coefficient of x^(r) in the expansion of (x+3)ⁿ⁻¹+(x+3)ⁿ⁻²(x+2)+(x+3)ⁿ⁻³(x+2)²+.+(x+2)ⁿ⁻¹ be α r. If Σ r=0 ⁿ α r =βⁿ-γⁿ, β, γ N, then the value of β²+γ² equals _ _ _ _ _.

Binomial Theorem and Its Simple Applications: JEE Main 2024 Mathematics question

JEE Main 2024 · 31 Jan, Evening shift · Q29

JEE Main 202431 Jan, Evening shiftMathematicsMediumBinomial Theorem and Its Simple Applications131× since 2002

Let the coefficient of

x^r

in the expansion of

(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots \ldots \ldots .+(x+2)^{n-1}

\alpha_r

. If

\sum_\limits{r=0}^n \alpha_r=\beta^n-\gamma^n, \beta, \gamma \in \mathbb{N}

, then the value of

\beta^2+\gamma^2

equals _________.

Your answerType your answer. Integers and decimals are accepted.