Question

Ten different letters of english alphabet are given. Out of these letters, words of 5 letters are formed. How many words are formed when at least one letter is repeated? (a) 69760 (b) 98748 (c) 96747 (d) 97147

Short answer

The number of 5-letter words with at least one letter repeated is 69,760.

Step-by-step solution

Calculate total number of 5-letter words without any restrictions

First, let's calculate the number of ways we can form 5-letter words without any restrictions. In such a case, we have 10 choices for the first letter, 10 choices for the second letter, 10 choices for the third letter, 10 choices for the fourth letter, and 10 choices for the fifth letter. Using the counting principle, the total number of 5-letter words without any restrictions is: \(10 \times 10 \times 10 \times 10 \times 10 = 10^5 = 100000\)

Calculate the total number of 5-letter words without any repetitions

In this case, we have 10 choices for the first letter, 9 choices for the second letter, 8 choices for the third letter, 7 choices for the fourth letter, and 6 choices for the fifth letter. Using the counting principle, the total number of 5-letter words without any repetitions is: \(10 \times 9 \times 8 \times 7 \times 6 = 30240\)

Calculate the number of words with at least one letter repeated

Now, we know the total number of words with no restrictions and the total number of words without any repetitions. To find the number of words with at least one letter repeated, we can subtract the number of words without any repetitions from the total number of words without any restrictions: Number of 5-letter words with at least one letter repeated = Total number of 5-letter words without any restrictions - Total number of 5-letter words without any repetitions \(= 100000 - 30240 = 69760\)

Identify the correct choice

The number of 5-letter words with at least one letter repeated is 69760. So the correct choice is: (a) 69760

Key concepts

Counting Principle

The Counting Principle is a fundamental rule in combinatorics, allowing us to calculate the total number of possible outcomes in a situation. It's like figuring out how many combinations you can make if you're layering options one after another. For example, if you have 10 options for picking your first choice, and for each of those choices you also have 10 options for a second choice, you multiply the choices together. In our exercise with letters, forming a 5-letter word without restrictions involves selecting each letter independently:

• 10 choices for the first letter,
• 10 choices for the second,
• 10 for the third, fourth, and fifth letters.

So, according to the counting principle, this gives us:\[ 10 \times 10 \times 10 \times 10 \times 10 = 10^5 = 100000 \] combinations.
This principle makes it straightforward to calculate the total potential outcomes in layered decision scenarios, simplifying complex combinatorial situations.

Permutations with Repetition

Permutations with repetition allow us to form arrangements where elements from a set can be repeated in the sequence. It happens when you're free to pick the same option multiple times.In the context of the problem, if we consider permutations with repetition, we're thinking about forming 5-letter words from 10 available letters, where letters can repeat. Because we can use any letter for any position, each of the 5 positions in the word has 10 options:

• The first letter has 10 options,
• so does the second, third, fourth, and fifth.

Thus, allowing letters to be repeated gives:\[ 10 \times 10 \times 10 \times 10 \times 10 = 100000 \]
permutations with repetition.
This concept helps explain how repetition broadens the scope of combinations drastically, favoring a larger number of possible outcomes.

Permutations without Repetition

When creating permutations without repetition, each choice affects the next, narrowing down future options since no option can be reused. This significantly changes the calculation approach.In our exercise, creating a 5-letter word without any repeated letters means:

• The first letter is chosen from 10 options,
• the second from the remaining 9,
• the third from 8,
• the fourth from 7,
• and the fifth from 6.

Thus, the calculation follows:\[ 10 \times 9 \times 8 \times 7 \times 6 = 30240 \]
permutations without repetition.
Understanding this concept shows how restrictions in selection can significantly limit the total possible outcomes, emphasizing the uniqueness of each arrangement.