Problem 49
Question
An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?
Step-by-Step Solution
Verified Answer
A voter can make a total of 20 different selections from the six candidates.
1Step 1: Understand the Formula for Combinations
The formula for calculating combinations is: \(C(n, r) = \frac{n!}{r!(n-r)!}\) where \(C(n, r)\) are the combinations, n is the total number of items, r is the number of items to choose, and '!' denotes a factorial.
2Step 2: Substitute into the Formula
Here, the number of candidates n is 6, and the number of city commissioners to select r is 3. Substitute these values into the combination formula: \(C(6, 3) = \frac{6!}{3!(6-3)!}\)
3Step 3: Simplify the Factorials
The factorials in the equation: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720, 3! = 3 x 2 x 1 = 6, and (6-3)! = 3! = 6.
4Step 4: Calculate the Combination
Replace the factorial values in the combination formula and calculate: \(C(6, 3) = \frac{720}{6 x 6} = 20\)
Other exercises in this chapter
Problem 49
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express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+
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Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$
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Verify the identity: $$\cot x+\tan x=\csc x \sec x$$
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