Arrangement of Coloured Flags

CAT 2000 Slot 1 · QA · Hard · Permutations and Combinations

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One red flag, three white flags and two blue flags are arranged in a line such that: I. No two adjacent flags are of the same colour II. The flags at the two ends of the line are of different colours

In how many different ways can the flags be arranged?

  1. A.

    6

  2. B.

    4

  3. C.

    10

  4. D.

    2

Answer

A

Explanation

There are 3 white flags (W), 2 blue flags (B), and 1 red flag (R), total 6 flags. To ensure no two adjacent white flags are together, white flags must occupy alternate positions (1, 3, 5) or (2, 4, 6). Case 1: W _ W _ W _ The empty slots are at 2, 4, 6. The remaining flags (2 B, 1 R) can be arranged in 3!2!=3\frac{3!}{2!} = 3 ways. Case 2: _ W _ W _ W The empty slots are at 1, 3, 5. The remaining flags (2 B, 1 R) can be arranged in 3!2!=3\frac{3!}{2!} = 3 ways. Total valid arrangements = 3+3=63 + 3 = 6.

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