If `f(x)=sum_(r=1)^(n) { r^(2) (""^(n)C_(r)- ^(n) C_(r-1))+ (2r+1) ^(n) C_(r)}`
and `f(30)=30(2)^(lambda),` then the value of `lambda` is

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the problem, we need to analyze the function \( f(x) \) given by:

\[ f(x) = \sum_{r=1}^{n} \left( r^2 \left( \binom{n}{r} - \binom{n}{r-1} \right) + (2r + 1) \binom{n}{r} \right) \]

We are also given that \( f(30) = 30 \cdot 2^{\lambda} \) and we need to find the value of \( \lambda \).

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