If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$,then the value of $r$ is equal to:

If all the letters of the word $'GANGARAM'$ are arranged,then the number of words in which exactly two vowels are together but no two $'G'$ occur together is-

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $x$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only two digits are repeated and each of these is repeated exactly twice. Then,

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?

In a group of $3$ girls and $4$ boys,there are two boys $B_1$ and $B_2$. The number of ways in which these girls and boys can stand in a queue such that all the girls stand together,all the boys stand together,but $B_1$ and $B_2$ are not adjacent to each other,is:

The number of non-negative integer solutions of the equations $6x + 4y + z = 200$ and $x + y + z = 100$ is