i) there are no restrictions:
nCr=r!(n−r)!n!
26C8=8!(26−8)!26!=8!×18!26!=1562275
ii) there must be 4 men and 4 women:
(16C4)(10C4)=(4!(16−4)!16!)(4!(10−4)!10!)
(16C4)(10C4)=(4!×12!16!)(4!×6!10!)=382200
iii) there should be an even number of women:
=(16C6)(10C2)+(16C4)(10C4)+(16C2)(10C6)+(16C0)(10C8)
=(6!(16−6)!16!)(4!(10−4)!10!)+(4!(16−4)!16!)(4!(10−4)!10!)+(2!(16−2)!16!)(6!(10−6)!10!)+(0!(16−0)!16!)(8!(10−8)!10!)
=(6!(10)!16!)(4!(6)!10!)+(4!(12)!16!)(4!(6)!10!)+(2!(14)!16!)(6!(4)!10!)+(0!(16)!16!)(8!(2)!10!)
=360360+382200+25200+45=767805
iv) there are more women than men:
=(16C0)(10C8)+(16C1)(10C7)+(16C2)(10C6)+(16C3)(10C5)
=(0!(16−0)!16!)(8!(10−8)!10!)+(1!(16−1)!16!)(7!(10−7)!10!)+(2!(16−2)!16!)(6!(10−6)!10!)+(3!(16−3)!16!)(5!(10−5)!10!)
=(!0(16)!16!)(8!(2)!10!)+(1!(15)!16!)(7!(1)!10!)+(2!(14)!16!)(6!(4)!10!)+(3!(13)!16!)(5!(5)!10!)
=45+1920+25200+141120=168285
v) there are atleast 6 men:
=(16C6)(10C2)+(16C7)(10C1)+(16C8)(10C0)
=(6!(16−6)!16!)(2!(10−2)!10!)+(7!(16−7)!16!)(1!(10−1)!10!)+(8!(16−8)!16!)(0!(10−0)!10!)
=(6!(10)!16!)(2!(8)!10!)+(7!(9)!16!)(1!(9)!10!)+(8!(8)!16!)(0!(10)!10!)
=360360+114400+12870=487630
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