Question

The number of permutations of $n\dots$ A... objects taken $r$ at a time, where $0\le r\le n$ and the objects do not repeat is ... ... which is denoted by ...C... . Here, A, B and C refer to

• A. different, $n(n-1)(n-2)\dots (n-r+1),{}^n{P}_r$
• B. similar, $n(n-1)\dots (n-r-1),{}^n{C}_r$
• C. different, $n(n-1)\dots (n-r+1),{}^n{C}_r$
• D. similar, $n(n-1)\dots (n-r+1){}^n{P}_r$

Answer 1

Answer: (A)

$A\to$ Different
$B\to n(n-1)(n-2)\dots (n-r+1)$
$C\to {}^n{P}_r$
There will be as many permutations as there are ways of filling in $r$ vacant places

by the $n$ objects. The first place can be filled in $n$ ways; following which, the second place can be filled in $(n-1)$ ways, following which the third place can be filled in $(n-2)$ ways,..., the $r^{th}$ place can be filled in $[n-(r-1)]$ ways. Therefore, the number of ways of filling in $r$ vacant places in succession is $n(n-1)(n-2)\dots .[n-(r-1)]$ or $n(n-1)(n-2)\dots (n-r+1)$.