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 colour are identical).

Answer 1

Total we have 12 flowers 3 red, 4 yellow and 5 white.
Number of arrangements $=\frac{12!}{3!4!5!}=27720.$
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 ${}^8{C}_5=56.$
$\therefore$ The number of ways for the $2^{nd}$ part $=35\times 56=1960.$