Question

the number of ways six people can be placed in a line for a photo can be determined using the expression 6!. what is the value of 6!? two of the six people are given responsibilities during the photo shoot. one person holds a sign and the other person points to the sign. the expression \\(\frac{6!}{(6 - 2)!}\\) represents the number of ways the two people can be chosen from the group of six. in how many ways can this happen? in the next photo, three of the people are asked to sit in front of the other people. the expression \\(\frac{6!}{(6 - 3)!3!}\\) represents the number of ways the group can be chosen. in how many ways can the group be chosen?

Explanation:

First Sub - Question: Find the value of \(6!\)

Step 1: Recall the definition of factorial

The factorial of a non - negative integer \(n\), denoted as \(n!\), is defined as \(n!=n\times(n - 1)\times(n - 2)\times\cdots\times1\) for \(n\gt0\) and \(0!=1\). For \(n = 6\), we have:
\(6!=6\times5\times4\times3\times2\times1\)

Step 2: Calculate the product

\(6\times5 = 30\), \(30\times4=120\), \(120\times3 = 360\), \(360\times2=720\), \(720\times1 = 720\)

Step 1: Simplify the denominator

First, calculate \(6-2 = 4\), so the denominator is \(4!\). We know that \(6!=6\times5\times4!\) (from the definition of factorial, \(n!=n\times(n - 1)!\))

Step 2: Substitute and simplify the fraction

\(\frac{6!}{4!}=\frac{6\times5\times4!}{4!}\)
The \(4!\) terms in the numerator and denominator cancel out, leaving \(6\times5=30\)

Step 1: Simplify the denominator terms

First, calculate \(6 - 3=3\), so the denominator is \(3!\times3!\). We know that \(6!=6\times5\times4\times3!\)

Step 2: Substitute and simplify the fraction

\(\frac{6!}{3!\times3!}=\frac{6\times5\times4\times3!}{3!\times3!}\)
Cancel out the \(3!\) in the numerator and denominator: \(\frac{6\times5\times4}{3\times2\times1}\)

Step 3: Calculate the numerator and denominator

The numerator \(6\times5\times4 = 120\), the denominator \(3\times2\times1=6\)

Step 4: Divide the numerator by the denominator

\(\frac{120}{6}=20\)

Answer:

\(720\)

Second Sub - Question: Evaluate \(\frac{6!}{(6 - 2)!}\)