In how many ways can a committee of $5$ members be formed from $4$ gentlemen and $6$ ladies such that the number of gentlemen is greater than the number of ladies?

In an examination,a question paper consists of $12$ questions divided into two parts,i.e.,Part $I$ and Part $II$,containing $5$ and $7$ questions,respectively. $A$ student is required to attempt $8$ questions in all,selecting at least $3$ from each part. In how many ways can a student select the questions?

If a question paper consists of $11$ questions divided into two sections,$I$ and $II$. Section $I$ consists of $6$ questions and section $II$ consists of $5$ questions,then the number of different ways a student can select $6$ questions,taking at least $2$ questions from each section,is

In how many ways can a committee be formed of $5$ members from $6$ men and $4$ women if the committee has at least one woman?

$A$ student has to answer $10$ out of $13$ questions in an examination,choosing at least $5$ questions from the first $6$ questions. The number of choices available to the student 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