Question

In how many ways can 6 persons stand in a queue?

Answer 1

The no. of ways in which 6 persons can stand in a queue is same as the number of arrangements of 6 different things taken all at a time, which is ordered pair is, do permutation, ${}^n{P_r}$

Complete step-by-step answer:
We are given a total of 6 persons standing in a queue. We need to find the no. of ways in which we can arrange these 6 persons to stand in the queue.

The no. of ways in which 6 persons can stand in a queue is the same as the no. of arrangements of 6 different things taken all the time. Now, this can be done through Permutation.

Now, the required no. of ways = ${}^6{P_6}$
Where, n = 6 and r = 6.
Thus, applying the formula, we get
$\begin{array}{l}^n{P_r}{\rm{ = }}{}^6{P_6}{\rm{ = }}\dfrac{{6!}}{{(6 - 6)!}}{\rm{ = }}\dfrac{{6!}}{{0!}}{\rm{ = }}\dfrac{{6!}}{1}\\\\\\\\{}^6{P_6}{\rm{ = 6! = 6 x 5 x 4 x 3 x 2 x 1}}\\\\\\\\{\rm{ = 720}}\\\\\end{array}$
Thus, there are 720 ways in which 6 persons can stand in a queue.

Note: You may confuse this question of permutation with that of ambition. If you do decombination, you get the answer as,
$\begin{array}{l}{}^n{C_r}{\rm{ = }}\dfrac{{n!}}{{(n - r)!{\rm{ r!}}}}\\\\{}^6{C_6}{\rm{ = }}\dfrac{{6!}}{{(6 - 6)!{\rm{ 6!}}}}{\rm{ = }}\dfrac{{6!}}{{0!{\rm{ 6!}}}}{\rm{ = 1}}\end{array}$
We get no. of arrangements as 1, which is not possible. Thus, use permutation.