Permutations an...

If $$^nC_{10} = ^nC_{12}$$, then find n?

A box contains $$2$$ white balls,$$3$$ black balls and $$4$$ red balls.In how many ways can three balls can be drawn from the box if atleast one black ball is to be included in the draw?

How many quadrilaterals can be formed by joining the vertices of an octagon?

The number of values of $$r$$ satisfying the equation $$\:^{69}C_{3r-1}-\:^{69}C_{r^2}=\:^{69}C_{r^2-1}-\:^{69}C_{3r}$$ is:

An event manager has ten patterns of chairs and eight patterns of tables. In how many ways can he make a pair of table and chair?

If $${ _{  }^{ n }{ C } }_{ 8 }={ _{  }^{ n }{ C } }_{ 5 }$$, then the value of $$n$$ is

When two coins are tossed and a cubical dice is rolled, then the total outcomes for the compound event is 

Simplify: $${ _{  }^{ 34 }{ C } }_{ 5 }+\sum _{ r=0 }^{ 4 }{ { _{  }^{ (38-r) }{ C } }_{ 4 } } $$.

If $$C\left( 28,2r \right) =C\left( 28,2r-4 \right) $$, then what is $$r$$ equal to?

Out of $$7$$ consonants and $$4$$ vowels, words are formed each having $$3$$ consonants and $$2$$ vowels. The number of such words that can be formed is