TITLE: Committee formation with Senators and Aides?
QUESTION [0 upvotes]: In a certain country, there are $100$ senators, each of whom has $4$ aides. These senators
and aides serve on various committees. A committee may consist either of $5$ senators,or
of $4$ senators and $4$ aides, or of $2$ senators and $12$ aides. Every senator serves on $5$
committees, and every aide serves on $3$ committees. How many committees are there
altogether?

REPLY [0 votes]: Let the number of the committees be: $C_{5,0}, C_{4,4}, C_{2,12}$
Every senator serves on 5 committees, and every aide serves on 3 committees, (I use double counting) $=> 5*100=5C_{5,0}+4C_{4,4}+2C_3, 3*400=0*4C_{5,0}+4C_{4,4}+12C_{2,12}$
Devide the second equation by 4 and add them up:
$500+300=5*(C_{5,0}+C_{4,4}+C_{2,12}) => C_{5,0}+C_{4,4}+C_{2,12}=160$ which is the total number of committees
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