The value of $\sum \limits_{r=0}^{22} {}^{22}C_{r} \cdot {}^{23}C_{r}$ is $.......$

How many words can be formed using $3$ consonants and $2$ vowels from $5$ consonants and $4$ vowels?

$A$ test containing $3$ objective-type questions is conducted in a class. Each question has $4$ options and only one option is the correct answer. No two students of the class have answered identically,and no student has written all correct answers. If every student has attempted all the questions,then the maximum possible number of students who have written the test is:

Three schools send $2, 4,$ and $6$ students,respectively,to a summer camp. The $12$ students must be accommodated in $6$ rooms numbered $1, 2, 3, 4, 5, 6$ in such a way that each room has exactly $2$ students and both are from the same school. The number of ways the students can be accommodated in the rooms is

In an admission test,there are $15$ multiple-choice questions. Each question has $4$ alternatives. For each question,one or more than one answer can be correct. If a student attempts all $15$ questions and marks the answers randomly,in how many different ways can the student answer the question paper?

The number of ways $5$ boys and $4$ girls can sit in a row so that either all the boys sit together or no two boys sit together,is $......$