Permutations an...

In the $$xy$$-plane, the vertices of a triangle are $$(-1,3), (6,3)$$ and $$(-1,-4)$$. The area of the triangle is ___ square units.

In how many ways $$7$$ men and $$7$$ women can be seated around a round table such that no two women can sit together?

Let $$\boxed { n }$$ be defined as $$\frac{(n+2)!}{(n-1)!}$$, what is the value of $$\frac{\boxed{7}}{\boxed {3}}$$ ?

Which of the following is correct?

Two vertices of a triangle are (2, 1) and (3, -2). Its third vertex is (x, y) such that $$\displaystyle y=x+3$$. If its area is 5 sq. units, what are the co-ordinates of the third vertex?

Twelve friends go out for a dinner to $$UTSAV$$ restaurant. There are two circular tables one with $$7$$ chairs and one with $$5$$ chairs. In how many ways can the group settle down themselves for dinner?

If $$n$$ books can be arranged on an ordinary shelf in $$720$$ ways, then in how many ways an these books be arranged in a circular shelf ? 

Area of the triangle formed by the points $$\left( 0,0 \right) ,\left( 2,0 \right) $$ and $$\left( 0,2 \right) $$ is 

Find the value of $$a$$ if area of the triangle is $$17$$ square units whose vertices are $$(0,0), (4,a), (6,4)$$.                         

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