Question

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

Answer 1

Numbers of ways of arrangement is determined by permutation.
Following is a short note to understand permutation and factorial in general:
A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.
The number of permutation of $n$ different objects taken $r$ at a time, where $0 < r \le n$and the objects do not repeat is $n(n - 1)(n - 2)...(n - r + 1)$, which is denoted by $P(n,r).$
$P(n,r) = \dfrac{{n!}}{{(n - r)!}}$
The notation $n!$represents the product of first $n$natural numbers,
i.e. the product $1 \times 2 \times 3 \times ... \times (n - 1) \times n = n!$.
$\begin{array}{l} 0! = 1\\\ 1! = 1\\\ 2! = 2 \times 1\\\ 3! = 3 \times 2 \times 1 \end{array}$

Complete step by step solution:
Step 1: Given that
Total numbers of person = 6
$\Rightarrow n = 6$
Step 2: find ‘r’
Numbers of persons standing in queue = 6
$\Rightarrow r = 6$
Step 3: find the numbers of ways or arrangement:
Numbers of ways 6 persons can stand in a queue = $P(n,r) = P(6,6)$
$P(n,r) = \dfrac{{n!}}{{(n - r)!}}$
$\begin{array}{l} = \dfrac{{6!}}{{(6 - 6)!}}\\\ = \dfrac{{6!}}{{0!}}\\\ = 6!\\\ = 6 \times 5 \times 4 \times 3 \times 2 \times 1\\\ = 720 \end{array}$

Therefore 6 persons can stand in a queue in 720 ways.

Note:
$n! = n(n - 1)!$
The number of permutations of $n$different things, taken r at a time, where repetition is allowed, is $\mathop n\nolimits^r$.
The number of permutations of $n$objects taken all at the time, where repetition is not allowed is $n!$.
Alternated steps:
Total numbers of persons =6
The number of permutation of 6 persons taken all at the time can stand in a queue $= 6!$
$= 720$
The number of permutation of $n$objects taken all at the time, where $\mathop p\nolimits_1$objects are of first kind, $\mathop p\nolimits_2$objects are of second kind, …, $\mathop p\nolimits_k$objects are of $\mathop k\nolimits^{th}$kind and rest, if any, are all different is
$\dfrac{{n!}}{{\mathop p\nolimits_1 !\mathop p\nolimits_2 !...\mathop p\nolimits_k !}}$