The number of ways to distribute $25$ apples among $4$ boys,such that each boy receives at least $4$ apples,is:

How many different signals can be transmitted by arranging $3$ red,$2$ yellow and $2$ green flags on a pole? [Assume that all the $7$ flags are used to transmit a signal.]

If the permutations of $A, B, C, D, E$ taken all together are written down in alphabetical order as in a dictionary and numbered,then the rank of the permutation $DEBAC$ is

In an examination,there are three multiple-choice questions,and each question has $4$ choices. The number of ways in which a student can fail to get all answers correct is:

All possible numbers are formed using the digits $1, 1, 2, 2, 2, 2, 3, 4, 4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is

$A$ box contains $2$ white balls,$3$ black balls and $4$ red balls. In how many ways can $3$ balls be drawn from the box,if at least one black ball is to be included in the draw?