In how many ways can $6$ apples be distributed among $3$ boys,there being no restriction to the number of apples each boy may get?

$\binom{n}{n-r} + \binom{n}{r+1}$,whenever $0 \le r \le n-1$,is equal to

The number of ways in which $5$ boys and $3$ girls can be seated in a row so that each girl is between two boys is:

Number of shortest paths from $HOSTEL$ to $ALLEN$ is equal to (as shown in the given figure):

The number of numbers between $2,000$ and $5,000$ that can be formed using the digits $0, 1, 2, 3, 4$ (repetition of digits is not allowed) which are multiples of $3$ is?

There are $16$ points in a plane,no three of which are in a straight line except $8$ which are all in a straight line. The number of triangles that can be formed by joining them equals