Number of ways of forming a committee of 6 members out of 5 Indians, 5 Americans and 5 Australians such that…

The number of ways a could form in which at least one member from each country in the committee.

We have triplets  $$\begin{gathered} \left(1,1,4\right)\to 3 \\ \left(1,2,3\right)\to 3!=6 \\ \left(2,2,2\right)\to 1 \end{gathered}$$

Now total ways

$=3\left({}^{5}C_1\times {}^{5}C_1\times {}^{5}C_4\right)+6\left({}^{5}C_1\times {}^{5}C_2\times {}^{5}C_3\right)+\left({}^{5}C_2\times {}^{5}C_2\times {}^{5}C_2\right)$

$=375+3000+1000=4375$