There are $5$ roads leading to a town from a village. The number of different ways in which a villager can go to the town and return back is:

There are $6$ men and $8$ women in a club. $A$ committee of $5$ people has to be formed. What will be the number of ways to select $5$ people if at least one woman is selected?

$A$ code word is to consist of $2$ English alphabets followed by $2$ distinct numbers between $1$ and $9$. For example,$CA23$ is a code word. How many such code words are there?

$^{n - 1}C_r = (k^2 - 3) \cdot ^nC_{r + 1}$ if $k \in$

In how many ways can five examination papers be arranged so that physics and chemistry papers never come together?

In how many different ways can the letters of the word '$LEADING$' be arranged in such a way that the vowels always come together?