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

Answer:

163

Step-by-step explanation:

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

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

2 judges-

$C^8_2=\dfrac{8!}{2!(8-2)!}=\dfrac{6!\cdot 7\cdot 8}{2\cdot 6!}=\dfrac{56}{2}=28$

ways to vote for 2 different judges

3 judges-

$C^8_3=\dfrac{8!}{3!(8-3)!}=\dfrac{5!\cdot 6\cdot 7\cdot 8}{2\cdot 3\cdot 5!}=\dfrac{6\cdot 56}{6}=56$

ways to vote for 3 different judges

4 judges-

$C^8_4=\dfrac{8!}{4!(8-4)!}=\dfrac{4!\cdot 5\cdot 6\cdot 7\cdot 8}{4!\cdot 4!}=\dfrac{5\cdot 6\cdot 7\cdot 8}{2\cdot 3\cdot 4}=5\cdot 7\cdot 2=70$

ways to vote for 4 different judges

In total, there are

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

different ways to vote.