Question

If permutation 1689 and combination 70. Find n and r ?

Answer 1

Given ${}^n{P}_r=1680\Rightarrow \frac{h!}{(h-r)!}=1680$
and, ${}^n{C}_r=70\Rightarrow \frac{n!}{r!(n-r)!}=70$
$\frac{1}{2}r!=\frac{\text{/}{1680}^{240}}{\text{/}{70}_{\text{/}10}}=24$
$\therefore r=4$
$\therefore \frac{n!}{(n-4)!}=1680$
$\Rightarrow (n-3)(n-2)(n-1)n=1680=2^4\times 3\times 7\times 5$
$=5\times 6\times 7\times 8$
$\therefore n=8$
$r=4$
(Take the combinations in such a way that you get consecutive numbers)
Hint : - It has to be around $7\&5$