In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

We may have (1 boy and 3 girls) or (2 boys and 2 girls) or (3 boys and 1 girl) or (4 boys). 

Required number of ways =  $6C1*4C3+6C2*4C2+6C3*4C1+6C4$  

=  $6C1*4C1+6C2*4C2+6C3*4C1+6C2$ = 209.