How many distinguishable code symbols can be formed with the letters for the word PHILOSOPHICAL?
13!/(2!2!2!2!2!)
194,594,400
Well, let me count the ways! The word "PHILOSOPHICAL" contains 13 letters. If we were just looking at distinct arrangements of these letters, without taking into account any repeated letters, we would have 13! (13 factorial) possibilities. However, since there are some repeated letters, we have to adjust for that.
Let's break it down, shall we? We have 2 "P's," 3 "I's," 1 "H," 1 "L," 1 "O," 2 "S's," 1 "C," 1 "A," and 1 "L." Now, to adjust for these repetitions, we divide the 13! by the factorials of the number of times each letter repeats.
So, the total number of distinguishable code symbols would be:
13! / (2! * 3! * 1! * 1! * 1! * 2! * 1! * 1!)
Now, if you'd like me to actually calculate that for you, I guess I could... but remember, it's a lot of math. And math is not typically known for its comedic value!
To determine the number of distinguishable code symbols that can be formed with the letters of the word PHILOSOPHICAL, we need to consider the total number of letters in the word and account for repetitions.
Step 1: Count the total number of letters.
The word PHILOSOPHICAL has 13 letters.
Step 2: Identify repeated letters.
In the word PHILOSOPHICAL, the letters 'H' and 'P' are repeated twice each.
Step 3: Calculate the number of distinguishable code symbols.
To find the number of distinguishable code symbols, we need to calculate the factorial of the total number of letters and divide it by the factorials of the repeated letters.
Total number of letters: 13
Repeated letter 'H' count: 2
Repeated letter 'P' count: 2
Distinguishable code symbols = 13! / (2! * 2!)
Calculating this expression:
13! = 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 6,227,020,800
2! = 2 * 1 = 2
2! = 2 * 1 = 2
Now substituting these values:
Distinguishable code symbols = 6,227,020,800 / (2 * 2) = 6,227,020,800 / 4 = 1,556,755,200
Therefore, there are 1,556,755,200 distinguishable code symbols that can be formed with the letters of the word PHILOSOPHICAL.
To find out how many distinguishable code symbols can be formed with the letters for the word PHILOSOPHICAL, we need to calculate the number of permutations.
A permutation is an arrangement of objects where the order matters. In this case, the objects are the letters of the word PHILOSOPHICAL.
The formula to calculate the number of permutations for a word with repeated letters is:
n! / (x1! * x2! * x3! * ...)
Where n is the total number of objects (in this case, the number of letters in the word) and x1, x2, x3, etc. represent the number of times each object is repeated.
Let's calculate the number of distinguishable code symbols for the word PHILOSOPHICAL:
1. Count the total number of letters in the word PHILOSOPHICAL. In this case, it is 13.
2. Identify the number of times each letter is repeated. Here's the breakdown:
- P: 1 occurrence
- H: 3 occurrences
- I: 2 occurrences
- L: 2 occurrences
- O: 2 occurrences
- S: 1 occurrence
- A: 1 occurrence
- C: 1 occurrence
3. Apply the formula n! / (x1! * x2! * x3! * ...). Substituting the values:
13! / (1! * 3! * 2! * 2! * 2! * 1! * 1! * 1!)
4. Simplify and calculate:
13! / (1 * 6 * 2 * 2 * 2 * 1 * 1 * 1) = 6,227,020
Therefore, there are 6,227,020 distinguishable code symbols that can be formed with the letters for the word PHILOSOPHICAL.