5P3/

12P4

5p^3/12p^4 =

5/12p

To understand how to solve these expressions, let's break them down.

5P3 represents the permutation of selecting 3 elements from a set of 5 elements. Permutation refers to the arrangement of objects in a specific order.

To calculate 5P3, you need to use the formula for permutations:

nPk = n! / (n - k)!

In this case, n represents the total number of elements in the set (5), and k represents the number of elements you want to select (3).

So, substituting the values into the formula:

5P3 = 5! / (5 - 3)!
= 5! / 2!

Now, let's calculate the factorials:
5! = 5 x 4 x 3 x 2 x 1 = 120
2! = 2 x 1 = 2

Substituting these values back into the equation:

5P3 = 120 / 2 = 60

Therefore, 5P3 is equal to 60.

Moving on to the second expression, 12P4, it represents the permutation of selecting 4 elements from a set of 12 elements.

Using the permutation formula, you can calculate 12P4 as follows:

12P4 = 12! / (12 - 4)!

Calculating the factorials:

12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 479,001,600
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

Substituting these values back into the equation:

12P4 = 479,001,600 / 40,320 = 11,880

Therefore, 12P4 is equal to 11,880.