The value of the expression ${ }^{47} C_4+\sum\limits_{j=1}^5{ }^{52-j} C_3$ is

The numbers $$1,2,3, \ldots \ldots, \mathrm{m}$$ are arranged in random order. The number of ways this can be done, so that the numbers $$1,2, \ldots \ldots ., \mathrm{r}(\mathrm{r}<\mathrm{m})$$ appears as neighbours is

Five balls of different colours are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is

$$ \text { If } 1000!=3^n \times m \text { where } m \text { is an integer not divisible by } 3 \text {, then } n= $$