Division and Distributuion into Groups of Unequal Sizes.

Q. Let $x_1{x}_2{x}_3{x}_4{x}_5{x}_6$ be a six digit number. The numbers of such numbers if
$x_1<x_2<x_3\le x_4<x_5<x_6$ is

1. ${}^9{C}_4$
2. ${}^9{C}_3$
3. ${}^{10}{C}_4$
4. ${}^{10}{C}_3$

Q. A committee of 5 is to be formed from 6 boys and 5 girls, . The number of ways that the committee can be formed so that the committee contains at least one boy and one girl is

1. 440
2. 445
3. 450
4. 455

Q. No. Of ways in which 12 identical balls can be put in 5 different boxes in a row, if no box remains empty is

1. ${}^{10}{C}_4$
2. ${}^{11}{C}_4$
3. ${}^{12}{C}_4$
4. ${}^9{C}_4$

Q. The total number of ways of dividing $15$ different things into groups of $8,4$ and $3$ respectively is

1. ${}^{15}{C}_8\cdot {}^7{C}_4$
2. ${}^{15}{C}_8\cdot {}^8{C}_4$
3. ${}^{15}{C}_6\cdot {}^7{C}_4$
4. ${}^{15}{C}_6\cdot {}^8{C}_4$

Q. Let $x_1{x}_2{x}_3{x}_4{x}_5{x}_6$ be a six digit number.The number of such numbers if
$x_1<x_2<x_3<x_4<x_5<x_6$ is

1. ${}^9{C}_3$
2. ${}^{10}{C}_3$
3. ${}^{10}{C}_6$
4. ${}^9{C}_5$

Q. The number of ways a pack of $52$ cards can be divided among four players in $4$ sets, three of them having $17$ cards each and the fourth one just $1$ card is :

1. $\frac{52!}{3!(17!{)}^3}$
2. $\frac{52!}{3\cdot (17!{)}^3}$
3. $\frac{52!}{(3!{)}^3(17!)}$
4. $\frac{52!}{(17!{)}^3}\times 4$