A pack contains n card numbered from 1 ton. Two consecutive numbered card are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k- 20 =....

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answer is D.

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Detailed Solution

We have,

$\begin{array}{l} 1 + 2 + 3 + \dots + n - 2 \leq 1224 \leq 3 + 4 + \dots + n \\ \Rightarrow \frac{(n - 1) (n - 2)}{2} \leq 1224 \leq \frac{(n - 2) (n + 3)}{2} \\ \Rightarrow n^{2} - 3 n - 2446 \leq 0 and n^{2} + n - 2454 \geq 0 \\ \Rightarrow 49 < n < 51 \\ \Rightarrow n = 50 \\ \Rightarrow \frac{n (n + 1)}{2} - (2 k + 1) = 1224 \\ \Rightarrow k = 25 \\ Thus, k - 20 = 25 - 20 = 5 \end{array}$

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