Question

You and five friends are posing for a photograph. In how many ways can you pose in a line for a photograph?

Answer 1

To find in how many ways you can pose in a line for a photograph given that there are 6 people (including you), we will assume that there are N people and N different positions. We have to arrange people in a line. We know that the first person can have any of the N places, the second person can have any of the (N-1) places and so on till the last person. To obtain the number of ways, we have to multiply all the outcomes we get and then substitute N as 6.

Complete step by step solution:
We need to find out how many ways you can pose in a line for a photograph given that there are 6 people (including you).
Let us consider that there are N people and N different positions. We can place first person in any one of the N places. Now, we have $\left( N-1 \right)$ places. We can place the second person in any of these $\left( N-1 \right)$ places. Therefore, the number of available places for the first two people can be written as $N\left( N-1 \right)$ .Now, we have $\left( N-2 \right)$ places. We can place the third person in $\left( N-2 \right)$ places. Hence, the number of available places for the first three people can be written as $N\left( N-1 \right)\left( N-2 \right)$ . We can do this till all the places are filled. Let us denote this as $N\times \left( N-1 \right)\times \left( N-2 \right)...\times 1=N!$ .
Now, we have to place six people in six positions. Similar to the above explained logic, we can do this in $6!$ ways.
$6!=6\times 5\times 4\times 3\times 2\times 1=720$
Hence, the answer is 720 ways.

Note: We can also denote the number of ways in which you can pose in a line of photograph, in terms of permutation. We can write it as $^{6}{{P}_{6}}$ . That is, there are 6 people and we have to place them in 6 positions. We know that $^{n}{{P}_{n}}=n!$ . Hence we will get 6! ways. The permutation formula is given as $^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$ . We use permutation here rather than combination because permutation is several ways of arranging few or all members within a specific order, whereas combination is a process of selecting the objects from a set or the collection of objects, such that the order of selection of objects does not matter. It refers to the combination of N things taken from a group of K at a time without repetition.