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Chapter 6: Problem 20

How many unordered sets are there of four items chosen from six?

Short Answer

Expert verified
There are \(15\) unordered sets of four items chosen from six.

Step by step solution

01

Understand the combination formula

The formula for calculating combinations is given by: \[C(n, r) = \frac{n!}{r!(n-r)!}\] where \(C(n, r)\) represents the number of combinations of selecting \(r\) items from a total of \(n\) items, and the ! symbol represents the factorial function (e.g., \(5! = 5 \times 4 \times 3 \times 2 \times 1\). In our problem, we have \(n = 6\) items and we want to choose \(r = 4\) items.
02

Apply the combination formula

Now that we know the formula, let's plug in the values for our problem: \[C(6, 4) = \frac{6!}{4!(6-4)!}\]
03

Calculate factorials

Calculate the factorials in the numerator and denominator: \[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\] \[4! = 4 \times 3 \times 2 \times 1 = 24\] \[(6-4)! = 2! = 2 \times 1 = 2\]
04

Substitute factorials into the formula

Now, let's substitute the factorials back into our formula and compute the number of combinations: \[C(6, 4) = \frac{720}{(24 \times 2)}\]
05

Calculate the number of unordered sets

Finally, calculate the number of unordered sets by dividing the numerator by the denominator: \[C(6, 4) = \frac{720}{48} = 15\] Therefore, there are 15 unordered sets of four items chosen from six.

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