Question

Messages are conveyed by arranging $4$ white, $1$ blue and $3$ red flags on a pole. Flags of the same colour are alike. If a message is transmitted 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

Answer 1

We will consider the following cases :

Case I - when $4$ alike and $2$ others alike flags are used $(4$ white and $2$ red ),
then no. of signals $=\frac{6!}{4!2!}=15$

Case II -when $4$ alike and $2$ others different flags are used $(4$ white, $l$ red and $1$ blue $)$ then no. of signals $=\frac{6!}{4!}=30$

Case III - when $3$ alike and $3$ others alike flags are used $(3$ white, $3$ red $)$ then no. of signals $=\frac{6!}{3!3!}=20$

Case IV - when $3$ alike and $2$ other alike and $1$ different flags are used $(3$ white, $1$ blue. $2$ red or $3$ red, $1$ blue, $2$ white$)$
then no. of signals $=2\times \frac{6!}{3!2!}=120$

Total $=15+30+20+120=185$