Ten people, including Fred, are in the Jazz Club. They decide to form a 3-person steering committee. How many possible committees can be formed that do not include Fred? Do you notice a relationship between your answers of the three parts of this problem?
+128474 Using my two previous answers.....
Total committees - Total committtees that include Fred = Total committees that don't include Fred
So we have
120 - 36 = 84 committees that don't include Fred
Note this is the same as C(9,3) = choosing any 3 of 9 people that don't include Fred
+128474 Using my two previous answers.....
Total committees - Total committtees that include Fred = Total committees that don't include Fred
So we have
120 - 36 = 84 committees that don't include Fred
Note this is the same as C(9,3) = choosing any 3 of 9 people that don't include Fred
+118609 total number of commities - no restrictions $${\left({\frac{{\mathtt{10}}{!}}{{\mathtt{3}}{!}{\mathtt{\,\times\,}}({\mathtt{10}}{\mathtt{\,-\,}}{\mathtt{3}}){!}}}\right)} = {\mathtt{120}}$$
total number that Fred is in $${\left({\frac{{\mathtt{9}}{!}}{{\mathtt{2}}{!}{\mathtt{\,\times\,}}({\mathtt{9}}{\mathtt{\,-\,}}{\mathtt{2}}){!}}}\right)} = {\mathtt{36}}$$
total number that Fred is not ine $${\left({\frac{{\mathtt{9}}{!}}{{\mathtt{3}}{!}{\mathtt{\,\times\,}}({\mathtt{9}}{\mathtt{\,-\,}}{\mathtt{3}}){!}}}\right)} = {\mathtt{84}}$$
$${\mathtt{84}}{\mathtt{\,\small\textbf+\,}}{\mathtt{36}} = {\mathtt{120}}$$
These 2 are complementary events,
either one happens OR the other happens.
