how many different 9-letter words (real or imaginary) can be formed from the letters in the word ECONOMICS?
nine letters, two sets of two.
9!/2!2!
check my thinking.
I still am in awe at what an imaginary word is....must be something in new age poetry.
imaginary as in mecosonic, just changing the order of the letters but more than likely it's not a word you'll see in the dictionary, that's what real or imaginary refers to.
To find the number of different 9-letter words that can be formed from the letters in the word ECONOMICS, we can use the concept of permutations.
Step 1: Count the number of each distinct letter in the word ECONOMICS:
- E: 1 time
- C: 2 times
- O: 2 times
- N: 1 time
- M: 1 time
- I: 1 time
- S: 1 time
Step 2: Calculate the total number of arrangements using the formula for permutations with repetition:
Total number of arrangements = (total number of letters)! / (number of times each letter repeats)!
In this case, the total number of letters is 9.
Number of arrangements = 9! / (1! * 2! * 2! * 1! * 1! * 1! * 1!) = (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (1 * 2 * 2 * 1 * 1 * 1 * 1 * 1 * 1)
= 362,880 / 4
= 90,720
Therefore, there are 90,720 different 9-letter words (real or imaginary) that can be formed from the letters in the word ECONOMICS.
To find the number of different 9-letter words that can be formed from the letters in the word "ECONOMICS," we can use the concept of permutations. A permutation is an arrangement of objects in a specific order. In this case, we want to find permutations of the 9 letters from "ECONOMICS."
There are three steps to solving the problem:
Step 1: Determine the number of letters we have (n = 9)
Step 2: Identify the number of repeating letters in the word "ECONOMICS."
- In this case, we have the letter 'O' repeated 3 times, the letter 'C' repeated 2 times, and the letter 'E' repeated 2 times.
- Let's denote the repetitions as r1 = 3, r2 = 2, and r3 = 2.
Step 3: Apply the formula for permutations with repetitions.
- The formula for permutations with repetitions is: P(n; r1, r2, ..., rn) = n! / (r1! * r2! * ... * rn!)
where "n!" represents the factorial of n, and "r1!" represents the factorial of r1, and so on.
Now, substituting the values into the formula:
P(9; 3, 2, 2) = 9! / (3! * 2! * 2!)
Calculating the factorial values:
9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880
3! = 3 × 2 × 1 = 6
2! = 2 × 1 = 2
Plugging in the values, we have:
P(9; 3, 2, 2) = 362,880 / (6 * 2 * 2) = 362,880 / 24 = 15,120
Therefore, there are 15,120 different 9-letter words (real or imaginary) that can be formed from the letters in the word "ECONOMICS."