A group consists of $$4$$ girls and $$7$$ boys. In how many ways can a team of $$5$$ members be selected is the team has at least one boy and one girl ?
Correct option is A. $$441$$
Total number of girls in the group $$=4$$
Total number of boys in the group $$=7$$
Probability of selecting atleast one boy and one girl =?
By considering following ways we can select
i) $$1$$ girls $$+4$$ boys $$\Rightarrow \ ^{4}C_{1}\times \ ^{7}C_{4}$$
ii) $$2$$ girls $$+3$$ boys $$\Rightarrow \ ^{4}C_{2}\times \ ^{7}C_{3}$$
iii) $$3$$ girls $$+2$$ boys $$\Rightarrow \ ^{4}C_{3}\times \ ^{7}C_{2}$$
iv) $$4$$ girls $$+1$$ boy $$\Rightarrow \ ^{4}C_{4}\times \ ^{7}C_{1}$$
Total number of ways $$=(\ ^{4}C_{1}\times \ ^{7}C_{4})+(\ ^{4}C_{2}\times \ ^{7}C_{3})+(\ ^{4}C_{3}\times\ ^{7}C_{2})+(\ ^{4}C_{4}\times \ ^{7}C_{1})$$
$$=140+210+84+7$$
$$=441$$