A signal is made by hoisting 7 flags one above the other on a pole. There are 3 red flags, 2 blue flags and 2 green flags; flags of the same colour are identical. If all 7 flags are used each time, how many different signals can be made?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 210
A signal is an arrangement of the 7 flags. Since flags of one colour are identical, swapping two of them gives the same signal, so \(7!\) is divided by the factorial of the number of flags of each colour.
Flags: 3 red, 2 blue, 2 green, 7 in all
Signals \(\displaystyle =\frac{7!}{3!\times2!\times2!}=\frac{5040}{6\times2\times2}=\frac{5040}{24}=210\)
Why the other options are incorrect
Option (2) 5040 — Incorrect. This is \(7!\), the count if all 7 flags were different.
Option (3) 420 — Incorrect. This is \(7!\div(3!\times2!)\), which treats one pair of same-coloured flags as different.
Option (4) 840 — Incorrect. This is \(7!\div3!\), which allows only for the red flags being identical.