If the permutations of $A, B, C, D, E$ taken all together are written down in alphabetical order as in a dictionary and numbered,then the rank of the permutation $DEBAC$ is

The number of permutations of $n$ things taken $r$ at a time,when $p$ things are always included,is

There are $7$ men and $8$ women in a club. $A$ committee is formed using these persons. What will be the number of ways to select $6$ persons such that at least one is a woman?

There are $n$ white and $n$ black balls marked $1, 2, 3, ..., n$. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours are

In how many ways can $5$ speakers $S_1, S_2, S_3, S_4$,and $S_5$ give speeches one after the other if $S_3$ wants to speak after $S_1$ and $S_2$?

On the sides $AB, BC, CA$ of a $\Delta ABC$,$3, 4, 5$ distinct points (excluding vertices $A, B, C$) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are: