How many words can be formed by taking $3$ consonants and $2$ vowels out of $5$ consonants and $4$ vowels?

$A$ test consists of $6$ multiple choice questions,each having $4$ alternative answers of which only one is correct. The number of ways in which a candidate answers all six questions such that exactly four of the answers are correct is:

The number of words that do not start and end with vowels,formed using all the letters of the word $UNIVERSITY$ such that all vowels are in alphabetical order,is

In a shop,there are $5$ types of ice-creams available. $A$ child buys $6$ ice-creams.
Statement-$1$: The number of different ways the child can buy the $6$ ice-creams is $^{10}C_5$.
Statement-$2$: The number of different ways the child can buy the $6$ ice-creams is equal to the number of different ways of arranging $6$ $A$'s and $4$ $B$'s in a row.

In an examination,there are three multiple-choice questions,and each question has $4$ choices. The number of ways in which a student can fail to get all answers correct is:

Number of shortest paths from $HOSTEL$ to $ALLEN$ is equal to (as shown in the given figure):