In how many ways can the letters of the word $TRIANGLE$ be arranged such that no two vowels occur together?

Seven scientists $S_1, S_2, \ldots, S_7$ are invited to deliver one lecture each in a conference. The number of ways all the seven lectures can be arranged such that the lecture of $S_1$ is prior to that of $S_3$ and the lecture of $S_3$ is prior to that of $S_7$ 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,$\frac{y}{9x} =$

Four speakers will address a meeting where speaker $Q$ will always speak before $P$. Then,the number of ways in which the order of speakers can be prepared is

How many words can be formed using the letters of the word $BHARAT$ such that $B$ and $H$ never come together?

How many numbers lying between $100$ and $1000$ can be formed with the digits $0, 1, 2, 3, 4, 5$,if the repetition of the digits is not allowed?