Question

Among the 10 nominees for two vacancies on City Council are 3 left-leaning, 4 right-leaning candidates, and 3 centralists. In how many ways can these vacancies be filled:a) With any two nomineesb) With any two of the right-leaning candidatesc) With one left-leaning and one right-leaning

Answer 1

To calculate in how many ways the vacancies can be filled, we'll be using combinations and permutations.

a) Any two candidates: C(10,2)

`\begin{gathered} C(10,2)=(10!)/(2!(10-2)!) \\ C(10,2)=(10\cdot9\cdot8!)/(2\cdot8!) \\ C(10,2)=5\cdot9 \\ C(10,2)=45 \end{gathered}`

Answer: 45 ways

b) Any two of the right-leading candidates: C(4,2)

`\begin{gathered} C(4,2)=(4!)/(2!(4-2)!) \\ C(4,2)=(4\cdot3\cdot2\cdot1)/(2\cdot2) \\ C(4,2)=3\cdot2=6 \end{gathered}`

Answer: 6 ways

c) With one right-leaning and one left-leaning: C(4,1)*C(3,1)

`\begin{gathered} C(4,1)=(4!)/(1!3!) \\ C(4,1)=(4\cdot3!)/(3!) \\ C(4,1)=4 \\ C(3,1)=(3!)/(2!) \\ C(3,1)=3 \\ C(4,1)\cdot C(3,1)=4\cdot3=12 \end{gathered}`

Answer: 12 ways