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Chapter 33: Problem 1063

Candidates for 3 different political offices are to be chosen from a list of 10 people. In how many ways may this be done?

Short Answer

Expert verified
There are 120 ways to choose candidates for the three different political offices from a list of ten people using the combination formula: \(C(10,3) = \frac{10!}{(3!(10-3)!)} = \frac{3,628,800}{(6 \times 5,040)} = 120\).

Step by step solution

01

Identify the number of items to choose from and the number of items to choose

In this problem, there are ten people to choose from (n = 10), and we need to choose three of them for three different political offices (r = 3).
02

Use the combination formula

The combination formula for choosing r items from a set of n items is given by: \[C(n,r) = \frac{n!}{(r!(n-r)!)}\] In this problem, n = 10 and r = 3. So, we will substitute these values into the combination formula to find the number of ways to choose 3 candidates from 10 people.
03

Calculate the factorials

Now, let's compute the factorials needed for the formula: \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800\) \(3! = 3 \times 2 \times 1 = 6\) \((10-3)! = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5,040\)
04

Substitute the factorial values into the combination formula

Next, we will substitute these factorial values into the combination formula and solve: \[C(10,3) = \frac{10!}{(3!(10-3)!)} = \frac{3,628,800}{(6 \times 5,040)}\]
05

Compute the final result

Now, we will compute the final result: \[C(10,3) = \frac{3,628,800}{(6 \times 5,040)} = \frac{3,628,800}{30,240} = 120\] There are 120 ways to choose candidates for the three different political offices from a list of ten people.

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