If the letters of the word $ATTEMPT$ are rearranged randomly,what is the probability that all $T$'s occur together?

An examination has $3$ multiple-choice questions,and each question has $4$ options. If a student passes only if they answer all questions correctly,in how many ways can they fail?

${ }^{34} C_5+\sum_{r=0}^4{ }^{(38-r)} C_4=$

Five boys and five girls are writing an examination in a hall in which $5$ benches are arranged in a row and only two students are to be seated on every bench at either of its ends. If the seating arrangement is to be such that no two boys or no two girls sit together as neighbours (a student should not have a student of same gender either on left or right),then the total number of such arrangements is

Let $5$-digit numbers be constructed using the digits $0, 2, 3, 4, 7, 9$ with repetition allowed,and are arranged in ascending order with serial numbers. Then the serial number of the number $42923$ is $...............$.

If the letters of the word $INTERMEDIATE$ are arranged,what is the probability that the two $E$'s do not come together?