Number of different words that can be formed from all letters of word $APPLICATION$ such that two vowels never come together is –

$n$ balls each of weight $w$ when weighted in pairs the sum of the weights of all the possible pairs is $120$ when they are weighed in triplets the sum of the weights comes out to be $480$ for all possible triplets, then $n$ is

If $P(n,r) = 1680$ and $C(n,r) = 70$, then $69n + r! = $

In how many ways can $6$ persons be selected from $4$ officers and $8$ constables, if at least one officer is to be included

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ element is :