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!?

Question

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 1

To find the value of \(6!\):

\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

So, the value of \(6!\) is 720.

Next, we will calculate the expression \(\frac{6!}{(6 - 2)!}\):

\[ \frac{6!}{(6 - 2)!} = \frac{6!}{4!} \]

Calculating \(4!\):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Now, substituting back into the expression:

\[ \frac{6!}{4!} = \frac{720}{24} = 30 \]

Thus, the number of ways to choose the two people from the group of six is 30.

Finally, we will find the number of ways to choose and arrange three of the six people sitting in front. For this, we use the expression \(\frac{6!}{(6 - 3)! , 3!}\):

\[ \frac{6!}{(6 - 3)! , 3!} = \frac{6!}{3! \times 3!} \]

As we already know \(6! = 720\) and \(3! = 6\):

\[ \frac{720}{3! \times 3!} = \frac{720}{6 \times 6} = \frac{720}{36} = 20 \]

So, the number of ways to choose the group of three people is 20.

In summary:

• The value of \(6!\) is 720.
• The number of ways to choose the two people is 30.
• The number of ways to choose the group of three people is 20.