13. A class of 22 students consists of 12 girls and 10 hoys. A committee of five

Question

13. A class of 22 students consists of 12 girls and 10 hoys. A committee of five studens is being chosen from this class to plan a school event. Dctermine the numher of 5 student committees that can be formed if: a the five students are chosen at random (1 mark) b. the committce must cunsist of 3 girls and 2 boys (1 mark) c. the committee must have 4 boys (1 mark) d. the committee must have 1 or 2 girls. (1 mark) c. Sam and Jordan must be on the committee, and the remaining students are randomly selected. (1 mark) f. there must be at least one boy on the committoe (1 mark)

Explanation / Answer

Number of girls=12
Number of boys=10
Total students in class=22
Number of students chosen for committe=5

Ans(a):
5 students out of 22 students can be chosen in C(22,5) ways.
NOTE: C(n,r) means combination of r items from n items.

So the final answer will be C(22,5)
=22!/(17!*5!)
=26334
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Ans(b):
3 girls out of 12 can be chosen in C(12,3) ways.
2 boys out of 10 can be chosen in C(10,2) ways.
So the final answer will be =C(12,3)*C(10,2)
=12!/(9!*3!)* 10!/(8!*2!)
=9900
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Ans(c):
must have 4 boys means there are two options
4boy+1girl or 5 boys
1 girls out of 12 can be chosen in C(12,1) ways.
4 boys out of 10 can be chosen in C(10,4) ways.

5 boys out of 10 can be chosen in C(10,5) ways.

So the final answer will be =C(12,1)*C(10,4) +C(10,5)
=12!/(11!*1!)* 10!/(6!*4!) + 10!/(5!*5!)
=12*210+252
=2772

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Ans(d):
must have 1 or 2 girls also means at least 1 girl must be there or you can say there can be 1 or more girls.
that can be found like
(all possible 5 member committee)-(all possible committee having all boys)
=C(22,5)-C(10,5)
=22!/(17!*5!)-10!/(5!*5!)
=26334-252
=26082

So the final answer will be = 26082
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Ans(e):
two seats for Sam and Jordan is fixed in 5 member committee so only 3 seats are vaccent for selection from remaining 20 students.
So the final answer will be = C(20,3)
=20!/(17!*3!)
=1140

So the final answer will be = 1140
---------------------------------

Ans(f):
must be at least 1 boy can be found like
(all possible 5 member committee)-(all possible committee having all girls)
=C(22,5)-C(12,5)
=22!/(17!*5!)-12!/(7!*5!)
=26334-792
=25542

So the final answer will be = 25542

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