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Applicants for the doctoral programmes of Ambi Institute of Engineering $\text{(AIE)}$ and Bambi Institute of Engineering $\text{(BIE)}$ have to appear for a Common Entrance Test $\text{(CET)}$. The test has three sections: Physics $\text{(P)}$, Chemistry $\text{(C)}$, and Maths $\text{(M)}$. Among those appearing for $\text{CET}$, those at or above the $80$th percentile in at least two sections, and at or above the $90$th percentile overall, are selected for Advanced Entrance Test $\text{(AET)}$ conducted by $\text{(AIE)}$. $\text{AET}$ is used by $\text{AIE}$ for final selection.

For the $200$ candidates who are at or above the $90$th percentile over all based on $\text{CET}$, the following are known about their performance in $\text{CET}$:

  1.  No one is below the $80$th percentile in all $3$ sections.
  2. $150$ are at or above the $80$th percentile in exactly two sections.
  3. The number of candidates at or above the $80$th percentile only in $\text{P}$ is the same as the number of candidates at or above the $80$th percentile only in $\text{C}$. The same is the number of candidates at or above the $80$th percentile only in $\text{M}$.
  4. Number of candidates below $80$th percentile in $\text{P}$: Number of candidates below $80$th percentile in $\text{C}$: Number of candidates below $80$th percentile in $\text{M}=4:2:1$.

$\text{BIE}$ uses a different process for selection. If any candidate is appearing in the $\text{AET}$ by $\text{AlE}$, $\text{BIE}$ considers their $\text{AET}$ score for final selection provided the candidate is at or above the $80$th percentile in $\text{P}$. Any other candidate at or above the $80$th percentile in $\text{P}$ in $\text{CET}$, but who is not eligible for the $\text{AET}$, is required to appear in a separate test to be conducted by $\text{BIE}$ for being considered for final selection. Altogether, there are $400$ candidates this year who are at or above the $80$th percentile in $\text{P}$.

If the number of candidates who are at or above the $90$th percentile overall and also at or above the $80$th percentile in all three sections in $\text{CET}$ is actually a multiple of $5$, then how many candidates were shortlisted for the $\text{AET}$ for $\text{AIE}?$

  1. $170$
  2. $169$
  3. $159$
  4. .$160$
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