How many possible ways to vote?

Question

There are 8 judges.
People can vote for 0, 1, 2, 3, or 4 people.
How many ways can a person vote?

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Answer 1 · 2018-05-21T00:21:51

163 ways.

There is $1$ way to vote for 0 people.

There are $8$ ways to vote for 1 person.

There are $(8*7)/2$ ways to vote for 2 people.

There are $(8*7*6)/(2*3)$ ways to vote for 3 people.

There are $(8*7*6*5)/(2*3*4)$ ways to vote for 4 people.

This is all because you can choose people but there are ways you can order the people. For example, there are $2*3$ ways to order the same 3 people.

Adding everything, we get $1+8+28+56+70=163$ .