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 comman 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 :

If all the letters of the word $'GANGARAM'$ be arranged, then number of words in which exactly two vowels are together but no two $'G'$ occur together is-

Let $S=\{1,2,3, \ldots ., 9\}$. For $k=1,2, \ldots \ldots, 5$, let $N_K$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$

A group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has at least $3$ girls $?$

A person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$, then $n$ equals

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 $.........$.