How many $5$-digit even numbers can be formed using the digits $1, 2, 5, 5, 4$?

There are $(n + 1)$ white and $(n + 1)$ black balls,each set numbered $1$ to $(n + 1)$. The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colours is

Find how many words can be formed out of the letters of the word '$ORIENTAL$' so that vowels always occupy the odd places.

The set $S = \{1, 2, 3, \ldots, 12\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \emptyset$. The number of ways to partition $S$ is

Let the eleven letters $A, B, \dots, K$ denote an arbitrary permutation of the integers $(1, 2, \dots, 11)$. Then, the product $(A - 1)(B - 2)(C - 3) \dots (K - 11)$ is:

To fill $12$ vacancies,there are $25$ candidates,of which $5$ are from the scheduled caste. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all,then the number of ways in which the selection can be made is: