Find where “M > R” does NOT necessarily hold true.

Introduction / Context:
The task is to identify the pattern under which the relation M > R is not guaranteed. Three options force M to be strictly greater than R; one leaves open the possibility that M equals R, breaking strictness.

Given Data / Assumptions:

• (a) M = P > Q > R
• (b) M > P ≥ Q = R
• (c) R = P < Q < M
• (d) R ≤ Q ≤ P = M (punctuation repaired for clarity)

Concept / Approach:
We check whether each option forces R strictly below M. Any scenario that allows R to equal M invalidates the strict “>” claim.

Step-by-Step Solution:
(a) With M = P and P > Q > R ⇒ M > R (forced).(b) Q = R and P ≥ R with M > P ⇒ M > R (forced).(c) R = P and P < Q < M ⇒ R < M (forced).(d) R ≤ Q ≤ P = M allows R = M (take R=Q=P=M). Then M > R is false. Hence (d) does not force M > R.

Verification / Alternative check:
Construct values for (d): let M=P=Q=R=5, which satisfies R ≤ Q ≤ P = M. Then M > R fails (5 > 5 is false).

Why Other Options Are Wrong:
In (a), (b), and (c), the chains require R to be strictly less than M, so M > R always holds.

Common Pitfalls:
Confusing “≥” with “>” and overlooking equality cases that break strict inequality. Always check whether equality is permitted somewhere on the path from R to M.

Final Answer:
R ≤ Q ≤ P = M