Cluster Pattern Identification

The task is to identify a letter cluster from the options that shares a pattern with the given examples: MDCL, QBAP, YHGX, TFES.

Analysis of Given Clusters

We assign numerical positions to each letter (A=1, Z=26) and examine the structure of the provided clusters:

• MDCL: M(13), D(4), C(3), L(12)
• QBAP: Q(17), B(2), A(1), P(16)
• YHGX: Y(25), H(8), G(7), X(24)
• TFES: T(20), F(6), E(5), S(19)

Two consistent rules emerge:

1. The fourth letter's position ($L_4$) is one less than the first letter's position ($L_1$). This can be expressed mathematically as:

   $ L_4 = L_1 - 1 $

   Examples: L(12) is 1 less than M(13); P(16) is 1 less than Q(17).
2. The third letter's position ($L_3$) is one less than the second letter's position ($L_2$). This can be expressed as:

   $ L_3 = L_2 - 1 $

   Examples: C(3) is 1 less than D(4); A(1) is 1 less than B(2).

Verification of Options

We test the given options against these two identified rules ($L_4 = L_1 - 1$ and $L_3 = L_2 - 1$):

• Option 1: XJIW (X=24, J=10, I=9, W=23)
  • Check Rule 1: $L_4 = L_1 - 1 \implies W(23) = X(24) - 1$. (True)
  • Check Rule 2: $L_3 = L_2 - 1 \implies I(9) = J(10) - 1$. (True)
  • This option satisfies both conditions.
• Option 2: FUVE (F=6, U=21, V=22, E=5)
  • Check Rule 1: $L_4 = L_1 - 1 \implies E(5) = F(6) - 1$. (True)
  • Check Rule 2: $L_3 = L_2 - 1 \implies V(22) = U(21) - 1$. (False, as $22 \neq 20$)
• Option 3: XCBY (X=24, C=3, B=2, Y=25)
  • Check Rule 1: $L_4 = L_1 - 1 \implies Y(25) = X(24) - 1$. (False, as $25 \neq 23$)
• Option 4: VDEW (V=22, D=4, E=5, W=23)
  • Check Rule 1: $L_4 = L_1 - 1 \implies W(23) = V(22) - 1$. (False, as $23 \neq 21$)

Final Answer Determination

Only Option 1, XJIW, consistently follows both established patterns ($L_4 = L_1 - 1$ and $L_3 = L_2 - 1$) observed in the initial letter clusters. Therefore, it is the correct answer.