The following observation is made about the scores obtained by 100 students in an exam: 'For each student, there exists another student in the class such that their scores are at most ten marks away.'

If the above statement is false, which one of the following statements is necessarily true?

Each one of the following clues contains a keyword that is partially filled.

Clue 1: Synonym of recognize (8 letters): _ D _ NT_ FY

Clue 2: A story long enough to fill a book (5 letters): $\_\_\_\_$

Clue 3: Two of something (6 letters): $\_\_\_\_$ PLE

Clue 4: A fraction of something, split equally into two parts (4 letters): $\_\_\_\_$ F

The first letter of each of the keywords can be rearranged to form a four-letter word. Which one of the options below is a possible choice for the four-letter word?

Three children P, Q, R and two grown-ups X, Y play a badminton doubles tournament. $X$ and $Y$ are parents to two of the children playing. The child of $X$ is not the same as the child of Y . Exactly one of the children does not have a parent playing in the tournament. The following rules are followed:

(i) A parent and his/her child cannot be on the same team.

(ii) A match can feature at most one parent and his/her child, that is, a maximum of one parent-child pair can play in a match.

$$ \text { The following matches were played: } $$

$$ \begin{array}{|l|c|c|} \hline & \text { TEAM 1 } & \text { TEAM2 } \\ \hline \text { MATCH 1 } & P \text { and } X & Q \text { and } R \\ \hline \text { MATCH 2 } & P \text { and } R & X \text { and } Y \\ \hline \text { MATCH 3 } & R \text { and } X & Q \text { and } Y \\ \hline \end{array} $$

Which one of the following options is correct?

Let $P_k$ represent the perimeter of a square with sides of length $k$. The value of the expression $P_1+P_2+P_3+\ldots+P_{10}$ is $\_\_\_\_$ .