Question

Messages are conveyed by arranging four white, one blue, and three red flags on a pole. Flags of the same color are alike. If a message is trasmitted by the order in which the colours are arranged, the total number of messages that can be transmitted if exactly six flags are used is

• A. $45$
• B. $65$
• C. $125$
• D. $185$

Answer 1

The correct option is D $185$

We will consider the following cases

Case	Flags	No. of signals
$4$ alike and $2$ others alike	$4$ white and $2$ red	$\frac{6!}{4!2!}=15$
$4$ alike and $2$ others different	$4$ white, $1$ red and $1$ blue	$\frac{6!}{4!}=30$
$3$ alike and $3$ other alike	$3$ white, $3$ red	$\frac{6!}{3!3!}=20$
$3$ alike and $2$ other alike and $1$ different	$3$ white, $1$ blue, $2$ red or $3$ red, $1$ blue, $2$ white	${}^2{C}_1\times \frac{6!}{3!2!}=120$
	Total	$185$