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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 StartFraction 6 factorial Over (6 minus 2) factorial EndFraction 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 StartFraction 6 factorial Over (6 minus 3) factorial 3 factorial EndFraction represents the number of ways the group can be chosen. In how many ways can the group be chosen?



Answer :

Answer:

1. Value of 6!

The number of ways six people can be placed in a line is given by 6! (6 factorial), which is calculated as: 6!=6×5×4×3×2×1=720

2. Number of ways two people can be chosen from six

To find the number of ways to choose 2 people from 6 for specific roles, we use the expression (6−2)!6! =4!6! =4!6×5×4! =6×5=30

So, there are 30 ways to choose 2 people from a group of 6 for specific roles.

3. Number of ways three people can be chosen from six

To find the number of ways to choose 3 people from 6 and arrange them in specific roles, we use the expression (6−3)!×3!6! : (6−3)!×3!6! =3!×3!6! =6×6720 =36720 =20

So, there are 20 ways to choose and arrange 3 people from a group of 

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