If ${}^nC_4, {}^nC_5$ and ${}^nC_6$ are in $A.P.$, then $n$ is

The value of ${ }^{34}C_5 + \sum_{i=0}^4 {}^{38-i}C_4$ is

The number of ways in which a cricket team of $11$ members can be formed out of $6$ batsmen,$6$ bowlers,$4$ all-rounders and $4$ wicket keepers by selecting at least $4$ batsmen,at least $3$ bowlers,at least $2$ all-rounders and only $1$ wicket keeper is:

$A$ committee of $4$ persons is to be formed from $2$ ladies,$2$ old men,and $4$ young men such that it includes at least $1$ lady,at least $1$ old man,and at most $2$ young men. The total number of ways in which this committee can be formed is:

There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?

$n$ balls each of weight $w$. When weighed in pairs,the sum of the weights of all possible pairs is $120$. When they are weighed in triplets,the sum of the weights of all possible triplets is $480$. Then $n$ is: