If $\binom{n+1}{3} = 2 \times \binom{n}{2}$,then $n = \dots$

The number of all possible ways of distributing $8$ identical apples among $3$ persons is

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

In how many ways can $5$ red and $4$ white balls be drawn from a bag containing $10$ red and $8$ white balls?

$A$ group of $9$ students,$s_1, s_2, \ldots, s_9$,is to be divided into three teams $X, Y$,and $Z$ of sizes $2, 3$,and $4$,respectively. Suppose that $s_1$ cannot be selected for team $X$,and $s_2$ cannot be selected for team $Y$. The number of ways to form such teams is: