Question

in an election 3 persons are to be elected from 6 candidates.A voter can cast any number of votes but not more than the candidates to be elected in how many was can he cast his vote

Answer 1

$$\begin{gathered} \mathrm{Since},\mathrm{a}\mathrm{voter}\mathrm{can}\mathrm{cast}\mathrm{any}\mathrm{number}\mathrm{of}\mathrm{votes}\mathrm{but}\mathrm{not}\mathrm{more}\mathrm{than}\mathrm{the}\mathrm{candidates}\mathrm{to}\mathrm{be} \\ \mathrm{elected}\mathrm{and}3\mathrm{persons}\mathrm{are}\mathrm{to}\mathrm{be}\mathrm{elected}\mathrm{from}6\mathrm{candidates}.\mathrm{Therefore}\mathrm{a}\mathrm{person}\mathrm{can} \\ \mathrm{cast}\mathrm{maximum}\mathrm{of}3\mathrm{votes}. \\ \mathrm{When}\mathrm{the}\mathrm{voter}\mathrm{casts}\mathrm{only}1\mathrm{vote},\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{to}\mathrm{cast}\mathrm{only}\mathrm{one}\mathrm{vote}={}^6\mathrm{C}_1=6 \\ \mathrm{When}\mathrm{the}\mathrm{voter}\mathrm{casts}2\mathrm{votes},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{to}\mathrm{cast}2\mathrm{votes}={}^6\mathrm{C}_2=15. \\ \mathrm{When}\mathrm{the}\mathrm{voter}\mathrm{casts}3\mathrm{votes},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{to}\mathrm{cast}3\mathrm{votes}={}^6\mathrm{C}_3=20. \\ \mathrm{Therefore},\mathrm{the}\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{a}\mathrm{voter}\mathrm{can}\mathrm{cast}\mathrm{a}\mathrm{vote}=6+15+20=31. \end{gathered}$$