If $^{n + 1}C_3 = 2 \cdot ^nC_2$,then $n =$

If $\binom{189}{35} + \binom{189}{x} = \binom{190}{x}$,then $x = \dots$

$A$ bag contains $3$ coins of one rupee,$4$ coins of fifty paise,and $5$ coins of ten paise. If at least one coin is selected from the bag,what is the total number of ways to make the selection?

If $\binom{18}{15} + 2\binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3}$,then find the value of $n$.

Compute $\frac{8!}{6! \times 2!}$

In an election,the number of candidates is $1$ more than the number of persons to be elected. If the voters can cast their votes in $254$ ways,find the number of candidates. ($A$ voter cannot cast more votes than the number of persons to be elected.)