If $1 \times 1 !+2 \times 2 !+3 \times 3 !+\ldots+n \times n !=11 !-1$, then the maximum value of ${ }^n C_r$ is

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Aicharya · Accepted Answer

$\begin{aligned} & 1 \times 1 !+2 \times 2 !+\ldots+n \times n ! \ & =(2-1) \times 1 !+(3-1) \times 2 !+\ldots+[(n+1)-1] \times n ! \ & =\{2 \times 1 !+3 \times 2 !+4 \times 3 !+\ldots(n+1) \times n !\} \ & \quad-\{1 \times 1 !+1 \times 2 !+1 \times 3 !+\ldots+1 \times n !\} \ & =(n+1) !-1 !=11 !-1 ! \quad \quad \text { (given) } \end{aligned}$ So, $n=10$ Now, maximum value of ${ }^{10} C_r$ occurs when $r=\frac{n}{2}=\frac{10}{2}=5$ Now, ${ }^{10} C_5=\frac{10 !}{5 !(10-5) !}=252$

If \(1 \times 1 !+2 \times 2 !+3 \times 3 !+\ldots+n \times n !=11 !-1\), then the maximum value of \({ }^n…

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