Question

A city council with eight members must elect a three-person committee. How many committees are possible?

Answer 1

The number of possible committees that can be formed from a city council with eight members is 56.

Step-by-step explanation:

To find the number of possible committees, we need to use the concept of combinations. Since there are 8 members in the city council and we need to choose a 3-person committee, we can use the formula for combinations:

nCr = n! / (r! * (n-r)!)

Substituting the values, we get:

nCr = 8! / (3! * (8-3)!) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56

Therefore, there are 56 possible committees.

Answer 2

This is a basic permutation and combination problem.

With permutations,

we care about the order of the elements.

With combinations,

we don't care about ordering.

In this problem, we don't really care in the order we select 3 people from 8 person. So, we will use combination. The answer is:

8 C 3.

The combination formula is:

`^nC_r=(n!)/(r!(n-r)!)_{}`

Note: n! is n * (n-1) * (n-2) * ...

Let's calculate 8C3. Shown below:

`\begin{gathered} ^nC_r=(n!)/(r!(n-r)!)_{} \\ ^8C_3=(8!)/(3!(8-3)!)_{} \\ =(8!)/(3!\cdot5!) \\ =(8\cdot7\cdot6\cdot5!)/(3\cdot2\cdot1*5!) \\ =(8\cdot7\cdot6)/(3\cdot2\cdot1) \\ =(8\cdot7\cdot6)/(6) \\ =8\cdot7 \\ =56 \end{gathered}`

There are 56 different committees possible.