a We want to find how many ways we can arrange all of the letters in the word PYRAMID. In this scenario, the order is important so we need to find the number of permutations of those 7 letters. We will use the Fundamental Counting Principle.
The number of possible choices for the next letter is decreased by one every time we choose a letter. Finally, we can multiply all of these numbers of choices to obtain the number of permutations.
Number of Permutations
7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
Therefore, there are 5040 ways we can arrange all of the letters in the word PYRAMID.
b When arranging 5 letters of the word PYRAMID, we begin the same way as we did in Part A. We still have 7 choices for the first letter, 6 choices for the second letter, and so on. But, this time, we will stop with the fifth letter. Let's use the Fundamental Counting Principle to find the number of permutations once again.
Number of Permutations
7 * 6 * 5 * 4 * 3 = 2520
When we want to arrange 5 letters of the word PYRAMID, we have 2520 ways.
