At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are $10$ candidates and $4$ are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote 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-

Words of length $10$ are formed using the letters, $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated ; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9 x}=$

If $^{43}{C_{r – 6}} = {\,^{43}}{C_{3r + 1}},$ then the value of $r$ is

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw