73) Find The No Of Arrangements In The Letter Of The Word CONGRATULATIONS It The Letter A Must Be Plated Next To Each Other. (teave Your Answer In Terms Of Foor Ctorial).

Feb 27, 2025 by ADMIN 174 views

73) Find the Number of Arrangements in the Letters of the Word CONGRATULATIONS with the Letter "A" Placed Next to Each Other

In this problem, we are tasked with finding the number of arrangements in the letters of the word CONGRATULATIONS with the letter "A" placed next to each other. This is a classic problem in combinatorics, and we will use the concept of permutations to solve it.

The word CONGRATULATIONS has 13 letters: C-O-N-G-R-A-T-U-L-A-T-I-O-N-S. We are asked to find the number of arrangements in these letters with the letter "A" placed next to each other. This means that the two "A"s must be treated as a single unit, and we will count the number of arrangements of this unit along with the remaining 11 letters.

To solve this problem, we will break it down into smaller sub-problems. First, we will count the number of arrangements of the 11 letters (excluding the two "A"s) and the single unit of "AA". Then, we will count the number of ways to arrange the two "A"s within the unit.

We have 11 letters (excluding the two "A"s) and a single unit of "AA". We can arrange these 12 units in 12! ways. However, we need to account for the fact that the two "A"s are identical, so we will divide by 2! to avoid overcounting.

Within the unit of "AA", we can arrange the two "A"s in 2! ways. However, since the two "A"s are identical, we will divide by 2! to avoid overcounting.

Now, we can calculate the total number of arrangements by multiplying the number of arrangements of the 11 letters and the "AA" unit by the number of ways to arrange the two "A"s within the unit.

Using the formula for permutations, we can calculate the total number of arrangements as follows:

12! / 2! * 2! = 12! / 4 = 479,001,600 / 4 = 119,750,400

Therefore, the total number of arrangements in the letters of the word CONGRATULATIONS with the letter "A" placed next to each other is 119,750,400.

In this problem, we used the concept of permutations to find the number of arrangements in the letters of the word CONGRATULATIONS with the letter "A" placed next to each other. We broke down the problem into smaller sub-problems and used the formula for permutations to calculate the total number of arrangements. The final answer is 119,750,400.
73) Find the Number of Arrangements in the Letters of the Word CONGRATULATIONS with the Letter "A" Placed Next to Each Other: Q&A

In our previous article, we solved the problem of finding the number of arrangements in the letters of the word CONGRATULATIONS with the letter "A" placed next to each other. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the problem.

A: Permutations refer to the number of ways to arrange a set of objects in a specific order. In the context of the problem, we are arranging the letters of the word CONGRATULATIONS.

A: When we have identical objects, such as the two "A"s in the word CONGRATULATIONS, we need to divide by the factorial of the number of identical objects to avoid overcounting. In this case, we divide by 2! to account for the fact that the two "A"s are identical.

A: Yes, the concept of permutations with repeated letters can be applied to other words with repeated letters. The formula for permutations with repeated letters is n! / (r1! * r2! * ... * rk!), where n is the total number of letters and r1, r2, ..., rk are the number of repeated letters.

A: When there are multiple repeated letters, we need to divide by the product of the factorials of the number of repeated letters. For example, if we have a word with three "A"s and two "T"s, we would divide by 3! * 2! to account for the repeated letters.

A: Yes, the concept of permutations can be applied to solve other problems involving arrangements. For example, we can use permutations to find the number of ways to arrange a set of objects in a specific order, or to find the number of ways to arrange a set of objects with certain restrictions.

A: Permutations have many real-world applications, such as:

Cryptography: Permutations are used to create secure encryption algorithms.
Computer Science: Permutations are used in algorithms for sorting and searching data.
Statistics: Permutations are used to calculate probabilities and test hypotheses.
Engineering: Permutations are used to design and optimize systems.

In this Q&A article, we provided additional insights and clarification on the problem of finding the number of arrangements in the letters of the word CONGRATULATIONS with the letter "A" placed next to each other. We also discussed the concept of permutations and its applications in various fields.