Question

There are 8 people on the ballot for regional judges. Voters can vote for any 4. Voters can choose to vote for 0¸ 1¸ 2¸ 3¸ or 4 judges. In how many different ways can a person vote?

Answer 1

163

Explanation:

0 judges -
`C^8_0=1` way to vote

1 judge -
`C^8_1=8` ways to vote

2 judges-

`C^8_2=(8!)/(2!(8-2)!)=(6!\cdot 7\cdot 8)/(2\cdot 6!)=(56)/(2)=28`

ways to vote for 2 different judges

3 judges-

`C^8_3=(8!)/(3!(8-3)!)=(5!\cdot 6\cdot 7\cdot 8)/(2\cdot 3\cdot 5!)=(6\cdot 56)/(6)=56`

ways to vote for 3 different judges

4 judges-

`C^8_4=(8!)/(4!(8-4)!)=(4!\cdot 5\cdot 6\cdot 7\cdot 8)/(4!\cdot 4!)=(5\cdot 6\cdot 7\cdot 8)/(2\cdot 3\cdot 4)=5\cdot 7\cdot 2=70`

ways to vote for 4 different judges

In total, there are

`1+8+28+56+70=163`

different ways to vote.