Generating Four-Letter Combinations from Mixed Letters
Understanding how to generate meaningful four-letter words, or combinations, from a given set of mixed letters can greatly enhance one's knowledge of permutations and combinations. This guide will walk through the methodology and calculations required to determine all possible four-letter words that can be formed from the set of letters provided: LZBONPINTLKC.
Introduction
The process involves breaking down the task into manageable sub-cases, each with specific constraints. By addressing each case in isolation, we can ensure a comprehensive approach to solving the problem and accurately determine the total number of four-letter words.
Total Unique Letters
The letters in the given string are: L Z B O N P I T K C. This gives us a total of 10 unique letters: L, Z, B, O, N, P, I, T, K, C.
Cases to Consider
Case 1: All Letters Are Different
In this scenario, we need to select 4 different letters each time we form a word.
Step 1: Determine the number of ways to choose 4 unique letters from the 10 available. This can be calculated as follows:
nCr(10, 4) 210
Step 2: Each selected group of 4 letters can be arranged in 4! (factorial) ways.
4! 24
Total for this Case:
210 * 24 5040
Case 2: One Letter Appears Twice and Two Other Letters Are Different
Here, we select one letter to appear twice and two other different letters from the remaining 9 letters.
Step 1: Choose one letter to appear twice. There are 2 options, L or N.
2 options
Step 2: Choose 2 different letters from the remaining 9 letters (excluding the one chosen to appear twice).
nCr(9, 2) 36
Step 3: Arrange the letters in the format AABC (e.g., LLZB, NNOP).
(4! / 2!) 12
Sub Total for Each of L and N:
36 * 12 432
Total for Case 2:
432 432 864
Case 3: Two Letters Appear Twice
This case can only occur with L and N as they are the only letters with duplicates.
Step 1: Arrange the letters in the format AABB (e.g., LLNN, NNPP).
(4! / (2! * 2!)) 6
Total for this Case:
6
Final Total
To find the total number of four-letter words, we sum the totals from all cases:
Case 1: 5040 Case 2: 864 Case 3: 6Total: 5040 864 6 5910
Hence, the total number of four-letter words that can be formed from the letters in LZBONPINTLKC is 5910.
Additional Examples
Let's provide some additional examples to further illustrate this concept:
For the word ZINC, there are 5 combinations: ZINC, ZION, PINT, PINK, POCK, PLOT, PICK, PALL, LINC, LION, LINT, LINK, TOLL, TILL, TILL, TILL, TILL, TILL... There are a total of 12 letters available: L (2), N (2), B (1), C (1), I (1), K (1), O (1), P (1), T (1), Z (1).By applying the same methodology as described, we can determine the number of four-letter words that can be formed from this set:
Case Breakdown:
Case 1: Taking all four distinct letters (L, N, B, C) (4! / 4!) 24 combinations Case 2: Taking two letters appearing twice (L, N) and two distinct letters (B, C) (4! / (2! * 2!)) 6 combinations Case 3: Arranging all four distinct letters (L, N, B, C) (4! / 4!) 24 combinationsTotal: 24 6 24 54
Therefore, the total number of four-letter words that can be formed from the letters in ZINC... is 54.
Conclusion
Understanding the principles of permutations and combinations is vital for constructing words from a given set of letters. This guide provides a detailed breakdown of the process and examples to help solidify the concept. By mastering these techniques, one can easily generate and analyze all possible four-letter words from any set of mixed letters.