# . Which among the following is/are correct

A: If an operation can be performed in 'm' different ways and a second operation can be performed in 'n' different ways, then both of these operations can be performed in 'm×n' ways together.

B: The number of arrangements of n different objects taken all at a time is !.

C: The number of permutations of n different things taken r at a time, when each thing may be repeated any number of times is n.

D: The number of circular permutations of 'n' different things taken all at a time is , if clockwise and anticlockwise orders are taken as different.

Explanation:

A If one operation can be done in 'm' different ways and another can be done in 'n' different ways, then either of these operations can be done in 'm×n' different ways at the same time.

It is correct because there are 'n' ways to perform each operation in the first, therefore these 'n' operations sum up to  'm' times, which equals n × m, which is same as ' m × n '.

B The number of arrangements of n different objects taken all at a time is !.

It is correct because when n things are needed to be put in 'n' places, there are n ways to fill the first gap and n - 1 ways to fill the second gap, so it becomes (n) (n – 1) (n – 2) ...(3) (2) (1) = !.

C When every thing can be repeated any number of times, the number of permutations of 'n' separate things taken r at a time is 'n'.

It is incorrect because the number of permutations in such case is greater than 'n' or equal to it.

is the total of circular permutations of 'n' different objects taken all at once, if the clockwise and counterclockwise orders are considered distinct.

It is incorrect because the number of circular permutations will be (n – 1) ! when taken together if the clockwise and counterclockwise orders are considered distinct.