Permutations an...

Write down and simplify:
The $$5^{th}$$ term of $${ \left( \cfrac { { x }^{ \frac { 3 }{ 2 }  } }{ { a }^{ \frac { 1 }{ 2 }  } } -\cfrac { { y }^{ \frac { 5 }{ 2 }  } }{ { b }^{ \frac { 3 }{ 2 }  } }  \right)  }^{ 8 }$$.

The number of positive integer satisfying the inequality $$^{n + 1}C_{n} - {}^{n + 1}C_{n - 1} \leq 100$$ is

If PQRS is a convex quadrilateral with 3, 4, 5 and 6 points marked on side PQ, QR, RS and PS respectively. Then, the number of triangles with vertices on different sides is

The number of triangles that can be formed by choosing the vertices from a set of $$12$$ points of which $$7$$ points lie on a line is

Find the $$4^{th}$$ term of $${ \left( 9x-\cfrac { 1 }{ 3\sqrt { x }  }  \right)  }^{ 18 }$$.

The arithmetic mean of $${ _{  }^{ n }{ C } }_{ 0 },{ _{  }^{ n }{ C } }_{ 1 },....{ _{  }^{ n }{ C } }_{ n }\quad $$, is

If $$^{n}C_{r - 1} = 36$$ and $$^{n}C_{r} = 84$$, then

If $$\displaystyle \sum _{ k=0 }^{ 18 }{ \cfrac { k }{ { _{  }^{ 18 }{ C } }_{ k } }  } =a\sum _{ k=0 }^{ 18 }{ \cfrac { 1 }{ { _{  }^{ 18 }{ C } }_{ k } }  } $$, then the value of $$a$$ is

There are $$10$$ persons including $$3$$ ladies. A committee of 4 persons including atleast one lady is to be formed. The number of ways of forming such a committee is :

The number of ways of distributing $$8$$ identical balls in $$3$$ distinct boxes, so that none of the boxes is empty is