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:

Step1: Calculate the value of 6!

The factorial formula is \(n!=n\times(n - 1)\times\cdots\times1\). So, \(6!=6\times5\times4\times3\times2\times1 = 720\).

Step2: Calculate the value of \(\frac{6!}{(6 - 2)!}\)

First, \((6-2)!=4!=4\times3\times2\times1=24\). Then \(\frac{6!}{(6 - 2)!}=\frac{6\times5\times4!}{4!}=6\times5 = 30\).

Step3: Calculate the value of \(\frac{6!}{(6 - 3)!3!}\)

\((6 - 3)!=3!=3\times2\times1 = 6\), and \(6!=720\). So \(\frac{6!}{(6 - 3)!3!}=\frac{6\times5\times4\times3!}{3!\times3\times2\times1}=\frac{6\times5\times4}{3\times2\times1}=20\).

Answer:

720
30
20