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:

The correct answer is 2.

To find all valid configurations using exactly two colours, we must examine each possible two-colour pairing: {Red, Blue}, {Red, Green}, and {Blue, Green}, and check them against all three rules for every row and column in the 5×5 grid.

Rule Recap (for any row or column of 5):

Rule 1: Adjacent beads must be different colours.
Rule 2: At least one Green bead must appear between any two Blue beads.
Rule 3: At least one Blue and at least one Green must appear between any two Red beads.

Case 1: Only Red & Blue (no Green beads)

Rule 1 forces strict alternation. For a row of 5, the only possible patterns are:
   R B R B R  or  B R B R B

In R B R B R: The Blue beads sit at positions 2 and 4. Between them is only a Red bead — there is no Green bead separating the two Blue beads. This violates Rule 2.

In B R B R B: Blue beads are at positions 1, 3, and 5. Between positions 1 and 3 there is only Red; between 3 and 5 only Red — no Green separates any pair of Blue beads. This violates Rule 2.

∴ The pair {Red, Blue} is impossible.

Case 2: Only Red & Green (no Blue beads)

Rule 1 again forces strict alternation. The only row patterns are:
   R G R G R  or  G R G R G

In R G R G R: Red beads are at positions 1, 3, and 5. Between positions 1 and 3 there is a Green bead, but no Blue bead. Rule 3 requires both a Blue and a Green between any two Reds. This violates Rule 3.

In G R G R G: Red beads are at positions 2 and 4. Between them is only a Green — again, no Blue bead is present. This violates Rule 3.

∴ The pair {Red, Green} is impossible.

Case 3: Only Blue & Green (no Red beads)

Rule 1 forces strict alternation. The only row patterns are:
   B G B G B  or  G B G B G

Check Rule 2 for B G B G B: Blue beads are at positions 1, 3, 5. Between position 1 and 3 there is a Green bead ✓; between 3 and 5 there is a Green bead ✓. Rule 2 satisfied.

Check Rule 2 for G B G B G: Blue beads are at positions 2 and 4. Between them is a Green bead ✓. Rule 2 satisfied.

Check Rule 3: Since there are no Red beads at all, Rule 3 is vacuously satisfied — it places no constraint. ✓

∴ The pair {Blue, Green} is potentially valid. Now we must count how many full 5×5 grid configurations exist.

Counting Valid 5×5 Configurations with {Blue, Green}

Since every row must be either BGBGB or GBGBG, and every column must also alternate, we need both row-wise and column-wise alternation to hold simultaneously.

Consider the cell at Row i, Column j. For columns to alternate, the colour at (i, j) must differ from the colour at (i+1, j). This means consecutive rows must be opposites of each other at every column position — i.e., if Row 1 is BGBGB, Row 2 must be GBGBG, Row 3 must be BGBGB, and so on.

This gives exactly two valid grid arrangements:

Configuration 1 (Row 1 starts with Blue):

Row 1: B G B G B
Row 2: G B G B G
Row 3: B G B G B
Row 4: G B G B G
Row 5: B G B G B

Verifying all 5 columns: Column 1 = B,G,B,G,B ✓   Column 2 = G,B,G,B,G ✓   Column 3 = B,G,B,G,B ✓   Column 4 = G,B,G,B,G ✓   Column 5 = B,G,B,G,B ✓

Configuration 2 (Row 1 starts with Green):

Row 1: G B G B G
Row 2: B G B G B
Row 3: G B G B G
Row 4: B G B G B
Row 5: G B G B G

Verifying all 5 columns: Column 1 = G,B,G,B,G ✓   Column 2 = B,G,B,G,B ✓   Column 3 = G,B,G,B,G ✓   Column 4 = B,G,B,G,B ✓   Column 5 = G,B,G,B,G ✓

No other starting pattern is possible (starting with anything other than B or G in position 1 contradicts the two-colour constraint), and these two configurations are distinct. Therefore, there are exactly 2 valid configurations.

Final Answer: 2

Only the colour pair {Blue, Green} can form a valid two-colour configuration. The pairs {Red, Blue} and {Red, Green} both violate the rules. The {Blue, Green} pair yields exactly 2 valid complete arrangements of the 5×5 grid.