How many arrangements can be made of the letters of the word '$ARRANGEMENT$'?

There are $6$ items in column $A$ and $6$ items in column $B$. $A$ student is asked to match each item in column $A$ with an item in column $B$. How many possible (correct or incorrect) answers are there to this question?

The number of words formed from the letters of the word $RAJASTHAN$ by taking all the letters at a time in which vowels are alternate is:

There are $3$ sections in a question paper and each section contains $5$ questions. $A$ candidate has to answer a total of $5$ questions,choosing at least one question from each section. Then the number of ways,in which the candidate can choose the questions,is

If $^{n + 1}{C_3} = 2{\,^n}{C_2},$ then $n =$

Four-digit numbers are formed using the digits $1, 2, 3, 4$ (repetition is allowed). The number of such four-digit numbers divisible by $11$ is