Permutations an...

Robert was asked to made a $$5$$ digit number from the digits $$2$$ and $$4$$ such that first digit cannot be $$4$$. Find the number of such $$5$$ digit numbers that can be formed .

If A and B are the sums of odd and even terms respectively in the expansion of $$(x+a)^n$$,then $$(x+a)^{2n} -(x-a)^{2n}$$ is equal to:

PQRS is a quadrilateral having $$3, 4, 5, 6$$ points in PQ, QR, RS and SP respectively. The number of triangles with vertices on different sides is?

How many $$3$$-digit even numbers can be formed from the digits $$1, 2, 3, 4, 5, 6$$ if the digits can be repeated?

The value of $$\sum\limits_{r = 1}^{10} r .\frac{{{}^n{C_r}}}{{{}^n{C_{r - 1}}}}$$ is equal to

The number of $$5$$ digit telephone numbers having least one of their digits repeated is 

Number of ways in which $$7$$ green bottles and $$8$$ blue bottles can be arranged in a row if exactly $$1$$ pair of green bottles is side by side is (Assume all bottles to be a like except for the colour).

There are $$10$$ points in a plane of which no $$3$$ points are collinear and $$4$$ points are concylic. No. of different circles that can be drawn through atleast $$3$$ points of these given points is

The sides AB, BC, CA of a triangle ABC have $$3,4$$ and $$5$$ interior points respectively on them. The number of triangles that can be constructed using these points as vertices is-

$$\frac{{({C_0} + {C_1})({C_1} + {C_2})({C_2} + {C_3}).....({C_{n - 1}} + {C_n})}}{{{C_0} + {C_1}({C_2}......{C_n})}} = $$