Question

lorelei evaluates the expression \\( \frac { 1 2! } { ( 1 2 - 1 0 )! 1 0! } \\) to determine how many different groups of ten she can make out of twelve items. her solution:

1. subtract within parentheses and simplify: \\( \frac { 6! } { ( 2 )! 5! } \\)
2. expand: \\( \frac { 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 } { 2 \cdot 1 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 } \\)
3. divide out common factors: \\( \frac { 6 } { 2 \cdot 1 } \\)
4. because 6 divided by 2 - 1 is 3, there are 3 ways to choose the groups.

which statements describe loreleis solution? check all that apply.
\\( \square \\) her work is correct.
\\( \square \\) her answer is correct.
\\( \square \\) in step 1, the subtraction cannot be completed before the factorial of each number is calculated.
\\( \square \\) in step 1, 12! divided by 10! is not equivalent to 6! divided by 5!.
\\( \square \\) in step 3, the dividing out of common factors was performed incorrectly.
\\( \square \\) there are sixty - six ways to choose ten items from twelve.

Explanation:

Step1: Analyze the combination formula

The formula for combinations is \(C(n,r)=\frac{n!}{(n - r)!r!}\). Here \(n = 12\) and \(r=10\), so \(C(12,10)=\frac{12!}{(12 - 10)!10!}=\frac{12!}{2!10!}\).

Step2: Simplify the factorial expression

We know that \(n!=n\times(n - 1)\times\cdots\times1\). So \(12! = 12\times11\times10!\). Then \(\frac{12!}{2!10!}=\frac{12\times11\times10!}{2\times1\times10!}\).

Step3: Cancel out common factors

Cancel out the \(10!\) terms. We get \(\frac{12\times11}{2\times1}=\frac{132}{2}=66\).

Step4: Evaluate Lorelei's steps

• In step 1: \(12!=(12\times11\times10!)\), so \(\frac{12!}{10!}=12\times11

eq6!\). Also, \((12 - 10)!=2!\), so the correct first - step after substitution is \(\frac{12!}{2!10!}\), not \(\frac{6!}{2!5!}\).

• In step 2: The expansion is wrong based on the incorrect first - step.
• In step 3: The division of common factors is wrong because the initial expression was wrong.
• The correct number of ways is \(66\) (using the combination formula \(C(12,10)=C(12,2)=\frac{12!}{2!(12 - 2)!}=\frac{12\times11}{2\times1}=66\))

Answer:

• In step 1, \(12!\) divided by \(10!\) is not equivalent to \(6!\) divided by \(5!\).
• In step 3, the dividing out of common factors was performed incorrectly.
• There are sixty - six ways to choose ten items from twelve.