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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 ?

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Solution

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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 $$=(\ ^{4}C_{1}\times \ ^{7}C_{4})+(\ ^{4}C_{2}\times \ ^{7}C_{3})+(\ ^{4}C_{3}\times\ ^{7}C_{2})+(\ ^{4}C_{4}\times \ ^{7}C_{1})$$

$$=140+210+84+7$$

$$=441$$

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