Division and Distribution into Groups of Equal Sizes

Q.

In a group of $6$ boys and $4$ girls, four children are to be selected. In how many ways can they be selected such that at least one boy should be there

1. $109$

2. $128$

3. $138$

4. $209$

Q. Five balls are to be placed in three boxes. Each box can hold all the five balls so that no box remains empty.

If balls as well as boxes are identical but boxes are kept in a row then number of ways is

Q. A rectangle with side lengths as $2m-1$ and $2n-1$ units is divided into squares of unit length by drawing parallel lines as shown in diagram, then the number of rectangles possible with odd side length is

1. $(m+n-1{)}^2$
2. $4^{m+n-1}$
3. $m^2{n}^2$
4. $m(m+1)n(n+1)$

Q. A rectangle with side lengths as $2m-1$ and $2n-1$ units is divided into squares of unit length by drawing parallel lines as shown in diagram, then the number of rectangles possible with odd side length is

1. $(m+n-1{)}^2$
2. $4^{m+n-1}$
3. $m^2{n}^2$
4. $m(m+1)n(n+1)$