How many words can be formed from the letters of the word $COURTESY$,whose first letter is $C$ and the last letter is $Y$?

If $a, b$ and $c$ are the greatest values of $^{19}C_{p}, ^{20}C_{q}$ and $^{21}C_{r}$ respectively,then

$A$ question paper has $2$ parts,part $A$ and part $B$,each containing $10$ questions. If a student has to choose $8$ questions from part $A$ and $5$ questions from part $B$,in how many ways can the student choose the questions?

In how many different ways can the letters of the word $MATHEMATICS$ be arranged?

In a group of $6$ boys and $4$ girls,four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

Let $A$ and $B$ be two sets containing $4$ and $2$ elements respectively. Then the number of subsets of the set $A \times B$,each having at least $3$ elements is: