Problem 26 - Algebra and Trigonometry

Question

Making Words A monkey is trained to arrange wooden blocks in a straight line. He is then given six blocks showing the letters \(A, E, H, L, M, T .\) What is the probability that he will arrange them to spell the word HAMLET?

Step-by-Step Solution

Answer

The probability is \( \frac{1}{720} \).

Step 1: Determine the Total Number of Arrangements

First, calculate the total number of ways to arrange six distinct letters. Since the letters A, E, H, L, M, and T are all unique, the total number of arrangements is the factorial of 6, which is 6!.

Step 2: Calculate 6!

Calculate 6! which represents the total number of unique arrangements possible. \[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\] This means there are 720 possible ways to arrange the letters.

Step 3: Identify the Favorable Outcomes

The favorable outcome is the arrangement where the letters are ordered to spell the word 'HAMLET'. There is exactly 1 way to spell 'HAMLET' with these six letters.

Step 4: Calculate the Probability

The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. Here, the probability is:\[ P(\text{HAMLET}) = \frac{1}{720} \] This is the probability that the monkey randomly arranges the letters to spell 'HAMLET'.

Key Concepts

Factorial: Understanding the Basics

When solving probability problems, the factorial is a frequently used mathematical concept. A factorial, denoted by an exclamation mark (!), refers to the product of all positive integers less than or equal to a particular number. For example, if we look at 6!, it means:

• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

This product gives us 720. In our exercise, this tells us that there are 720 different ways to arrange six unique blocks.

Factorials are handy whenever you need to figure out how many different sequences you can form with a set of items when all items are different. It's like having a set of books and wondering how many ways you can line them up on a shelf.

Remember, as the number of items increases, the factorial grows very quickly. So, for seven items, you calculate 7! which would be a much larger number.

Permutations: Arranging Different Items

In situations where the order of arrangement matters, like spelling a word, permutations come into play. A permutation refers to a specific sequence or arrangement of a set of items.

• For the word 'HAMLET', we're interested in finding out all possible sequences formed by arranging the letters A, E, H, L, M, and T.

Since we're dealing with six distinct letters, the total number of permutations would be given by 6!, or 720 as calculated.

Permutations differ from combinations, where order doesn't matter. In this exercise, the sequence "HAMLET" is different from "TALMEH", thus making them unique permutations. Essentially, anytime you reshuffle the arrangement of a set of items, you're generating a new permutation.

That's the beauty of permutations—providing insight into how order changes meaning in sequences.

Favorable Outcomes: Finding the Desired Result

When calculating probability, understanding favorable outcomes is crucial. A favorable outcome is simply the event or result that we are interested in. In our example, the favorable outcome is arranging the blocks to form the word 'HAMLET'.

• There is precisely one way to achieve this because once the letters are in the order 'HAMLET', none of the letters can move.

The total number of arrangements we found using the factorial and permutations techniques is 720. Thus, in this scenario, our favorable outcomes are limited to one, as the exact arrangement of the blocks in the order 'HAMLET' is sought.

The probability is subsequently calculated by comparing the number of favorable outcomes (1 way to spell 'HAMLET') with the total potential outcomes (720 permutations), resulting in the probability of \[P( ext{HAMLET}) = \frac{1}{720}\]

Recognizing favorable outcomes helps in determining how likely it is for a particular event to occur among many possibilities.