Permutations an...

How many $$3$$ digit numbers can we make using the digits $$4,5,7$$ and $$9$$ and where repetition is allowed?

A password for a computer system requires exactly $$6$$ characters. Each character can be either one of the $$26$$ letters from A to Z or one of the ten digits from $$0$$ to $$9$$. The first character must be a letter and the last character must be a digit. How many different possible passwords are there?

The exponent of $$5$$ in $$^{120}C_{60}$$, is

If $$^nC_{r-1}=36, ^nC_r = 84$$ and $$^nC_{r+1} = 126$$ then the value of $$^nC_8$$ is:

The value of the expression $$^{47}C_4$$ +$$ \sum_{j=1}^{5} { }^{52-j}C_3$$ is

If $$^{n-1}C_r = (k^2 - 3) ^nC_{r+1} ,$$ then k belongs to the interval

If $$^{20}C_r = ^{20}C_{r-10}$$, then the value of $$^{18}C_r$$ is:

The exponent of $$7$$ in $$^{120}C_{50}$$,  is

For a chess tournament $$13$$ people were selected for quarter finals. Each person plays two matches with the other. How many matches have been held in the whole tournament?

The area of regular polygon of n sides with length of side $$\sqrt{3}$$, where n > 1 and satisfies the relation $$\displaystyle \sum_{r=0}^{n} \frac{n^2-3n+3}{2\ ^nC_r}=\sum_{r=0}^n\frac{r}{^nC_r}$$ is :