Question

Calculate the number of permutations of the letters \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\) taken two at a time.

Short answer

There are 12 permutations of the letters a, b, c, and d taken two at a time, calculated using the formula P(n, r) = n! / (n - r)!, with n = 4 and r = 2.

Step-by-step solution

Identify the values of n and r

In this problem, we have 4 objects (letters a, b, c, and d) and we want to arrange 2 of them at a time. So, n = 4 and r = 2.

Use the formula for permutations

Now, we will use the formula for permutations, P(n, r) = n! / (n - r)!, where n = 4 and r = 2.

Calculate the factorial values

We need to calculate the factorial values for n and n - r: n! = 4! = 4 × 3 × 2 × 1 = 24 and (n - r)! = (4 - 2)! = 2! = 2 × 1 = 2

Substitute the factorial values into the formula

Now, substitute the calculated values into the formula: P(4, 2) = 24 / 2

Calculate the number of permutations

Finally, calculate the number of permutations: P(4, 2) = 12 There are 12 permutations of the letters a, b, c, and d taken two at a time.