the number of ways six people can be placed in a line for a photo can be determined using the expression 6…

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!?\ntwo 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?\nin 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?

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!?\ntwo 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?\nin 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?

Answer

Explanation:

Step1: Calculate 6!

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

Step2: Calculate (\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 (\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