$\binom{n}{n-r} + \binom{n}{r+1}$,whenever $0 \le r \le n-1$,is equal to

In a city,no two persons have an identical set of teeth,and there is no person without a tooth. Also,no person has more than $32$ teeth. If we disregard the shape and size of the teeth and consider only the presence or absence of a tooth at each of the $32$ positions,then the maximum population of the city is:

How many different ways can the letters in the word $ATTEND$ be arranged?

If $x, y$ and $r$ are positive integers, then $^x{C_r} + ^x{C_{r-1}} \cdot ^y{C_1} + ^x{C_{r-2}} \cdot ^y{C_2} + \dots + ^y{C_r} = $

The number of integers greater than $6000$ that can be formed using the digits $3, 5, 6, 7,$ and $8$ without repetition is:

We are to form different words with the letters of the word $INTEGER$. Let $m_1$ be the number of words in which $I$ and $N$ are never together and $m_2$ be the number of words which begin with $I$ and end with $R$,then $m_1/m_2$ is equal to