Question

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

Short answer

There are 24 different ways to arrange the four letters a, b, c, and d, taken four at a time. This was determined by using the permutation formula \( \frac{{n!}}{{(n-r)!}} \), where \( n \) represents the total number of items (4), and \( r \) represents the number of items chosen at a time (4).

Step-by-step solution

Understand the problem

In this problem, we want to find out how many ways we can arrange 4 letters - a, b, c, and d. Because the order of arrangement matters here, we are dealing with permutations.

Identify values for n and r

In this case, \( n \), the total number of items, is 4 (a, b, c, and d). And \( r \), the number of items to choose at a time, is also 4 (since we're taking four at a time).

Plug into the formula

Now that we've identified \( n \) and \( r \), we will substitute them into the permutation formula \( \frac{{n!}}{{(n-r)!}} \). This gives us \( \frac{4!}{(4-4)!} \)

Simplifying the Equation

Evaluate 4! (factorial), which is the product of all positive integers from 1 to 4. This gives us 4 x 3 x 2 x 1 = 24. Now we perform (4-4)! which equals 0! According to factorial rules, any 0! equals 1 (by definition).

Final Calculation

Divide 24 by 1, which gives us 24. So, there are 24 different ways to arrange the four letters a, b, c, and d.