Question:medium

Twenty five coloured beads are to be arranged in a grid comprising of five rows and five columns. Each cell in the grid must contain exactly one bead. Each bead is coloured either Red, Blue or Green.
While arranging the beads along any of the five rows or along any of the five columns, the rules given below are to be followed:
(1) Two adjacent beads along the same row or column are always of different colours.
(2) There is at least one Green bead between any two Blue beads along the same row or column.
(3) There is at least one Blue and at least one Green bead between any two Red beads along the same row or column.
Every unique, complete arrangement of twenty five beads is called a configuration.
The total number of possible configuration using beads of only two colours is:
[This Question was asked as TITA]

Updated On: Jan 15, 2026
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The Correct Option is B

Solution and Explanation

To resolve this issue, we must examine how to generate two-color arrangements under the specified conditions. Given that only two of the three available colors (Red, Blue, Green) will be utilized, each possible color pairing will be analyzed.
  1. Red and Green Combination:
    • Rule 1 (adjacent beads differ): This is achievable by alternating Red and Green in every row and column.
    • Rule 2 (Green between two Blues): This rule is inapplicable as Blue is not part of this arrangement.
    • Rule 3 (Blue and Green between Reds): This rule is also inapplicable due to the absence of Blue.
    • Result: A valid arrangement is produced by a consistent alternating pattern of Red and Green, such as a checkerboard.
  2. Blue and Green Combination:
    • Rule 1: Similar to the above, alternate Blue and Green in rows and columns.
    • Rule 2: This rule is inherently satisfied as Green is adjacent to all Blues.
    • Rule 3: This rule is inapplicable as Red is not included.
    • Result: An alternating pattern (e.g., checkerboard) fulfills the requirements.
  3. Red and Blue Combination:
    • Rule 1: Achievable through alternating bead placement.
    • Rule 2: Requires Green between Blues, but Green is unavailable.
    • Rule 3: Requires Green between Reds, but Green is unavailable.
    • Result: Violates Rules 2 and 3, thus no valid arrangement exists.
Initially, two configurations are possible: one for Red-Green and one for Blue-Green. Therefore, the total number of valid configurations using only two colors is 2.
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