List all the permutations of five objects $x, y, z, s$, and $t$ taken two at a time without repetition. What is ${ }_{5} \mathrm{P}_{2}$ ? List all the permutations of five objects $x, y, z, s$, and $t$ taken two at a time without repetition. Choose the correct answer below. A. $x, y, z, s, t$ B. $x y, x z, x s, x t, y x, y z, y s, y t, z x, z y, z s, z t, s x, s y, s z, s t, t x, t y$, C. $x x, x y, x z, x s, x t, y y, y z, y s, y t, z z, z s, z t, s s, s t, t t$ D. $x y, x z, x s, x t, y z, y s, y t, z s, z t, s t$

Step 1 ：First, we need to understand the question. It asks for all the permutations of five objects $x, y, z, s$, and $t$ taken two at a time without repetition. This means we are looking for all possible pairs of these five objects, where the order of the objects in each pair matters and each object can only appear once in each pair.

Step 2 ：We can start by choosing the first object in the pair. There are 5 options: $x, y, z, s$, and $t$.

Step 3 ：After choosing the first object, we then choose the second object. Since we cannot repeat the first object, there are only 4 remaining options.

Step 4 ：Therefore, for each of the 5 options for the first object, there are 4 options for the second object. So, there are $5*4=20$ possible pairs in total.

Step 5 ：We can list all these pairs: $x y, x z, x s, x t, y x, y z, y s, y t, z x, z y, z s, z t, s x, s y, s z, s t, t x, t y, t z, t s$.

Step 6 ：This matches option B in the question.

Step 7 ：Next, we need to calculate ${ }_{5} \mathrm{P}_{2}$, which is the number of permutations of 5 objects taken 2 at a time. The formula for permutations is ${ }_{n} \mathrm{P}_{r} = n! / (n-r)!$, where $n$ is the total number of objects, $r$ is the number of objects taken at a time, and $!$ denotes factorial.

Step 8 ：Substituting $n=5$ and $r=2$ into the formula, we get ${ }_{5} \mathrm{P}_{2} = 5! / (5-2)! = 5*4*3*2*1 / 3*2*1 = 5*4 = 20$.

Step 9 ：So, ${ }_{5} \mathrm{P}_{2} = 20$, which matches the number of pairs we listed earlier.

Step 10 ：Therefore, the correct answer to the question is option B and ${ }_{5} \mathrm{P}_{2} = \boxed{20}$.