From a group of $7$ men and $6$ women,five persons are to be selected to form a committee so that at least $3$ men are there on the committee. In how many ways can it be done?

In how many different ways can the letters of the word '$OPTICAL$' be arranged so that the vowels always come together?

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

If ${ }^{2 n} C_{r} = { }^{2 n} C_{r+2}$,find $r$.

In a certain test,${a_i}$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, 3, ..., k$. No student gave more than $k$ wrong answers. The total number of wrong answers given is:

The set $S = \{1, 2, 3, \ldots, 12\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \emptyset$. The number of ways to partition $S$ is