Permutations an...

If $$p> q$$, the number of ways of $$p$$ men and $$q$$ women can be seated in a row so that no two women sit together is

Total number of arrangements of the letters of the word SUCCESS such that both $$C's$$ are together and no two $$S's$$ are together is

Out of $$15$$ persons $$10$$ can speak Hindi and $$8$$ can speak English. If two persons are chosen at random, then the probability that one person speaks Hindi only and the other speaks both Hindi and English is

A telephone number $$d_1d_2d_3d_4d_5d_6d_7$$ is called memorable if the prefix sequence $$d_1d_2d_3$$ is exactly the same as either of the sequence $$d_4d_5d_6$$ or $$d_5d_6d_7$$(or possibly both). If each $$d_1\epsilon\{x|0\leq x\leq 9, x\epsilon W\}$$, then number of distinct memorable telephone number is(are).

$$\sum _{ r=0 }^{ n-1 }{ \cfrac { { _{  }^{ n }{ C } }_{ r } }{ { _{  }^{ n }{ C } }_{ r }+{ _{  }^{ n }{ C } }_{ r+1 } }  } $$ is equal to

A man $$x$$ has $$7$$ friends, $$4$$ of them are ladies and $$3$$ are men. His wife $$Y$$ also has $$7$$ friends, $$3$$ of them are ladies and $$4$$ are men. Assume $$X$$ and $$Y$$ have no common friends. Then, the total number of ways in which $$X$$ and $$Y$$ together can throw a party inviting $$3$$ ladies and $$3$$ men , so that $$3$$ friends of each of $$X$$ and $$Y$$ are in this party, is 

The sum $$^{20}C_0+^{20}C_1+^{20}C_2+..... +^{20}C_{10}$$ is equal to

The value of $$\begin{pmatrix} 30 \\ 0 \end{pmatrix}\begin{pmatrix} 30 \\ 10 \end{pmatrix}-\begin{pmatrix} 30 \\ 1 \end{pmatrix}\begin{pmatrix} 30 \\ 11 \end{pmatrix}+\begin{pmatrix} 30 \\ 2 \end{pmatrix}\begin{pmatrix} 30 \\ 12 \end{pmatrix}-...+\begin{pmatrix} 30 \\ 20 \end{pmatrix}\begin{pmatrix} 30 \\ 30 \end{pmatrix}$$ is where $$\begin{pmatrix} n \\ r \end{pmatrix}={ _{  }^{ n }{ C } }_{ r }$$

Let $${ T }_{ n }$$ denotes the number of triangles which can be formed by using the vertices of a regular polygon of n sides. If $${ T }_{ n+1 }\ -\ { T }_{ n }=21$$, then $$n$$ is equal to

If $$15! =2^{\alpha}\cdot 3^{\beta}\cdot 5^{\gamma}\cdot 7^{\delta}\cdot 11^{\theta}\cdot 13^{\Phi}$$, then the value of expression $$\alpha -\beta +\gamma -\delta +\theta -\Phi$$ is