A signal is made by hoisting 8 flags one above the other on a pole. There are 4 red flags, 2 blue flags and 2 white flags; flags of the same colour are identical. If all 8 flags are used each time, how many different signals can be made?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 420
Flags of the same colour are alike, so divide \(8!\) by the factorial of the number of flags of each colour.
\(8!=40320\)
\(4!\times2!\times2!=24\times2\times2=96\)
Signals \(\displaystyle =\frac{40320}{96}=420\)
Why the other options are incorrect
Option (1) 840 — Incorrect. This is \(\frac{8!}{4!\times2!}\); it allows for only one of the two pairs of identical flags.
Option (2) 40320 — Incorrect. This is \(8!\), which treats every flag as different.
Option (3) 210 — Incorrect. This divides by 2! once too often; only blue and white come in pairs.