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Question

A committee of 10 is to be formed from 8 teachers and 12 students of whom 4 are girls. In how many ways this can be done, so that the committee contains atleast four of either groups (teachers and students) and atleast 2 girls and atleast 2 boys are in the committee.

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Solution

8teachersand12students(4girls+8boys)
Required:committeeof10withatleast4of
eithergroupsandatleast2girls
and2boys.
case(i):4students,6teachers
numberofwaysofselecting6teachers
outof8=86C
4studentsmustbesuchthat2aregirlsand
2boys.So,numberofways=42C.82C
numberofwaysofselecting=86C.42C.82C
case(ii):5students,5teachers
Numberofwaystoselect5teachers=85C
5studentscanbeeither3girls,2boys
so,Numberofways=85C(42C.82C+42C.83C)
case(iii):6students,4teachers
6studentscaneitherbe2girls,4boysor3girls,3boys
or4girls,2boys
Numberofwaystoselect4teachers=84C
So,numberofways=84C(42C.84C+43C.83C+44C.82C)
totalnumberofways=86C.42C.82C+85C(43C.82C+42C.83C)+84C(42C.84C+43C.83C+44C.82C)
=4704+56(4×28+6×56)+70(6×70+4×56+1×28)
=76,832ways

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