A country has ten smart cities. The government decides to connect all these cities by road. How many roads the government need to construct so that every city is connected to every other city ?

If for some $\mathrm{m}, \mathrm{n} ;{ }^6 \mathrm{C}_{\mathrm{m}}+2\left({ }^6 \mathrm{C}_{\mathrm{m}+1}\right)+{ }^6 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3$ and ${ }^{n-1} P_3:{ }^n P_4=1: 8$, then ${ }^n P_{m+1}+{ }^{n+1} C_m$ is equal to

The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets themselves such that they get consecutive blocks of $5, 3 $ and $2$ tickets is

The number of $4-$letter words, with or without meaning, each consisting of $2$ vowels and $2$ consonants, which can be formed from the letters of the word $UNIVERSE$ without repetition is $.........$.

In a shop there are five types of ice-creams available. A child buys six ice-creams.               

Statement $-1 :$ The number of different ways the child can buy the six ice-creams is $^{10}C_5.$ 

Statement $-2 :$ The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6 \,A's$ and $4 \,B's$ in a row.

If ${a_n} = \sum\limits_{r = 0}^n {} \frac{1}{{^n{C_r}}}$ then $\sum\limits_{r = 0}^n {} \frac{r}{{^n{C_r}}}$ equals