If ${ }^{1} \mathrm{P}_{1}+2 \cdot{ }^{2} \mathrm{P}_{2}+3 \cdot{ }^{3} \mathrm{P}_{3}+\ldots+15 \cdot{ }^{15} \mathrm{P}_{15}={ }^{\mathrm{q}} \mathrm{P}_{\mathrm{r}}-\mathrm{s}, 0 \leq \mathrm{s} \leq 1$ then ${ }^{\mathrm{q}+\mathrm{s}} \mathrm{C}_{\mathrm{r}-\mathrm{s}}$ is equal to .... .

The value of $\sum\limits_{r = 0}^{n - 1} {\frac{{^n{C_r}}}{{^n{C_r} + {\,^n}{C_{r + 1}}}}} $ equals

A car will hold $2$ in the front seat and $1$ in the rear seat. If among $6$ persons $2$ can drive, then the number of ways in which the car can be filled is

If $^{2017}C_0 + ^{2017}C_1 + ^{2017}C_2+......+ ^{2017}C_{1008} = \lambda ^2 (\lambda   > 0),$ then remainder when $\lambda $ is divided by $33$ is-

How many words, with or without meaning, each of $2$ vowels and $3$ consonants can be formed from the letters of the word $\mathrm{DAUGHTER}$ ?

A person wants to climb a $n-$ step staircase using one step or two steps. Let $C_n$ denotes  the number of ways of climbing the $n-$ step staircase. Then $C_{18} + C_{19}$ equals