The letter EQUATION without repetitond,how many 4 letter codes can be formed,if it contain 3 consonants

To solve the problem of determining how many 4-letter codes can be formed using the letters from "EQUATION" with no repetitions and containing exactly 3 consonants, we'll break it down step by step.

### Step 1: Identify the consonants and vowels in "EQUATION."
- **Consonants:** Q, T, N (3 consonants)
- **Vowels:** E, U, A, I, O (5 vowels)

### Step 2: Choose 3 consonants and 1 vowel.
- Since there are exactly 3 consonants (Q, T, N), the only way to choose 3 consonants is to use all of them.
- There are 5 vowels (E, U, A, I, O), so we can choose 1 vowel from these 5. The number of ways to choose 1 vowel from 5 is \( \binom{5}{1} = 5 \).

### Step 3: Arrange the chosen letters.
- Once the 4 letters (3 consonants and 1 vowel) are chosen, they can be arranged in \( 4! \) ways.
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

### Step 4: Calculate the total number of codes.
The total number of codes is the product of the number of ways to choose the vowel and the number of ways to arrange the 4 letters:

\[
\text{Total number of codes} = 5 \times 24 = 120
\]

### Final Answer:
There are **120** possible 4-letter codes that can be formed using the letters from "EQUATION," with no repetitions and containing exactly 3 consonants.

Would you like more details or have any questions?

---

Here are some related questions you might find interesting:

1. How many 4-letter codes can be formed using only vowels from "EQUATION"?
2. What if the 4-letter codes had to contain exactly 2 consonants and 2 vowels?
3. How many 5-letter codes can be formed from "EQUATION" with 3 consonants and 2 vowels?
4. How many 4-letter codes can be formed using all consonants from "EQUATION"?
5. What if repetitions were allowed in forming the 4-letter codes?

**Tip:** When solving combinatorial problems, carefully identifying and categorizing the available elements (like consonants and vowels) is crucial for accurate counting.

Discover how many 4-letter codes can be formed using the letters from EQUATION with 3 consonants and no repetitions. Learn about permutations and combinations in combinatorial mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation