Question

In how many ways we can arrange $3$ red flowers, $4$ yellow flowers and $5$ white flowers in a row$?$
In how many ways this is possible if the white flowers are to be separated in any arrangement$?$ (Flowers of same color are identical).

Answer 1

Total we have $12$ flowers $3$ red, $4$ yellow and $5$ white.
Number of arranqernents$=\frac{12!}{3!4!5!}=27720.$
For the second part, first arrange $3$ red & $4$ yellow
This can be done in $\frac{7!}{3!4!}=35$ ways
Now select $5$ places from among $8$ places (including extremes) & put the white flowers there. This can be done in ${}^8{\text{C}}_5=56.$
$\therefore$ The number of ways for the $2^{nd}$ part$=35\times 56=1960.$