There are $4$ oranges,$5$ apples,and $7$ mangoes in a fruit basket. The number of ways of selecting at least one fruit from among the fruits in the basket is

The least value of $n$ such that ${ }^{(n-1)} C_3 + { }^{(n-1)} C_4 > { }^n C_3$ is:

$A$ country has $10$ smart cities. The government decides to connect all these cities by road. How many roads does the government need to construct so that every city is connected to every other city?

$A$ committee of $5$ is to be formed out of $6$ men and $4$ ladies. The number of ways this can be done,when at most $2$ ladies are included,is

The number of ways in which $17$ apples can be distributed among $4$ guests such that each guest gets at least $3$ apples 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 $.........$.