Question

How many permutations exist of the letters a, b, c, d taken two at a time?

Answer 1

Answer: The correct option is (a) 12.

Step-by-step explanation:  We are given to find the number of permutations that exists for the letters a, b, c and d taking two at a time.

We know

The number of permutations of 'n' different things taking 'r' ('r' less than or equal to 'n') at a time is given by the formula:

$^nP_r=\dfrac{n!}{(n-r)!}.$

In the given case, there are 4 different letters and we are to take two at a time, so

n = 4  and  r = 2.

Therefore, the number of permutations will be

$^4P_2=\dfrac{4!}{(4-2)!}=\dfrac{4!}{2!}=\dfrac{4\times3\times2\times 1}{2\times 1}=4\times 3=12.$

Thus, there are 12 permutations that exists.

Option (a) is correct.

Answer 2

The number of permutations of the 4 different letters, taken two at a time, is given by:
$4P2=\frac{4!}{2!}=12$