A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl ? (ii) at least one boy and one girl ? (iii) at least 3 girls ?

Explanation

Number of girls = 4 and Number of boys = 7
We have to select a team of 5 members provided that
(i) team having on girls.
`therefore` Required selection `= ""^(7)C_(5) = (7"!")/(5"!" 2"!") = (7 xx 6)/(2) = 21`
(ii) atleast one boy and one girl
`therefore` Required selection `= ""^(7)C_(1) xx ""^(4)C_(4) + ""^(7)C_(2) xx ""^(4)C_(3) + ""^(7)C_(3 ) xx ""^(4)C_(2) + ""^(7)C_(4) xx ""^(4)C_(1)`
`7 xx 1 + 21 xx 4 + 35 xx 6 + 35 xx 4`
`=7 + 84 + 210 + 140 = 441`
(iii) when atleast three girls are included `= ""^(4)C_(3) + ""^(7)C_(2) + ""^(4)C_(4) xx ""^(7)C_(1 )`
`= 4 xx 21 + 7 = 84 + 7 = 91`