Q. 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 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______
Permutations and Combinations
Solution:
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 $3$ others 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
$^2C_1 \times \frac{6!}{3!2!} = 120$
Total
$185$
| 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 $3$ others 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 | $^2C_1 \times \frac{6!}{3!2!} = 120$ |
| Total | $185$ |