Question

If each person can serve on at most two committees.

Answer 1

Case $1.$ Each person serves at only one committee we have to choose $3$ from $6={}^6{C}_3=20$,

but they can be chosen for any committee. Total $=3!\times 20$

Total $=120.$

Case $2.$ One person serves in $2$ committees. That person can be any of the $6={}^6{C}_1=6.$

Person chosen for the reamining comm $={}^5{C}_1=5.$

But he can be chosen for any committee total $=6\times 5\times 3=90.$

$\Rightarrow$ Sum $=120+90=210.$

Hence, the answer is $210.$