Sequences and S...

The sides of a quadrilateral are all positive integers and three of them are $$5, 10, 20.$$ How many possible value are there for the fourth side?

In a test there were n questions. In the test $$\displaystyle 2^{n-i}$$ students gave wrong answers to i questions where $$\displaystyle i=1,2,3...,n$$. If the total number of wrong answers given is 2047 then n is

If $$^{19}C_r$$ and $$^{19}C_{r-1}$$ are in the ratio 2:3, then find $$^{14}C_r$$

If $$^nC_3=^nC_{13}$$, then $$^{20}C_n$$ is.

If the difference of the number of arrangements of three things from a certain number of dissimilar things and the number of selections of the same number of things from them exceeds $$ 100$$, then the least number of dissimilar things is

Determine the area of the triangle whose vertices are $$\displaystyle \left( \frac { 1 }{ 2 } ,\frac { -1 }{ 2 }  \right) ,\left( 2,\frac { -1 }{ 2 }  \right) $$ and $$\displaystyle \left( 2,\frac { \sqrt { 3 } -1 }{ 2 }  \right) $$.

A college offers $$7$$ courses in the morning and $$5$$ courses in the evening. Find the number of ways a student can select exactly one course either in the morning or in the evening.

If $$m$$ denotes the number of $$5$$ digit numbers if each successive digits are in their descending order of magnitude and $$n$$ is the corresponding figure. When the digits and in their ascending order of magnitude then $$(m-n)$$ has the value

There are $$2$$ identical white balls, $$3$$ identical red balls and $$4$$ green balls of different shades. The number of ways in which they can be arranged in a row so that atleast one ball is separated from the balls of the same colour, is

If j, k, and n are consecutive integers such that $$0 < j < k < n$$ and the units (ones) digit of the product jn is 9, what is the units digit of k ?