I got this question wrong on my test, and I have to correct it :/
How many permutations are there of all the letters in the word "TEXAS" ?
The answer to this part is 120, but I have no idea on how to solve the second part without having to write it all down.
*How many of these permutations begin with 2 consonants?
Permutations: 5!
Permutations with beginning consonants:
3*4!
Wait... so it's 3 multiplied by 4? Sorry, I'm not used to reading when it's typed out.
3 multiplies by 4 factorial.
To find the number of permutations that begin with 2 consonants in the word "TEXAS," we can break it down into steps:
Step 1: Count the number of consonants in the word "TEXAS." In this case, the consonants are "T," "X," and "S." So, there are 3 consonants.
Step 2: Determine the number of ways we can choose and arrange 2 consonants at the beginning. Since we want to start with 2 consonants, we need to select 2 out of the 3 consonants. We use a combination to calculate this.
The notation for combination is nCr, where n represents the total number of items and r represents the number of items to be chosen. In this case, we have 3 consonants, and we want to choose 2 of them.
3C2 = 3! / (2! * (3-2)!) = 3! / (2! * 1!) = 3
So, there are 3 ways to choose and arrange 2 consonants at the beginning.
Step 3: Determine the number of permutations of the remaining letters. After selecting the 2 consonants, we are left with 4 letters: "E," "A," "S," and "T." We need to find the number of permutations for these 4 letters, which is the same as finding the number of arrangements of these letters.
The formula to calculate permutations is n!, where n represents the number of items to be arranged. In this case, we have 4 letters, so the number of permutations is 4!.
4! = 4 x 3 x 2 x 1 = 24
Step 4: Multiply the results from steps 2 and 3 to find the total number of permutations that begin with 2 consonants.
3 x 24 = 72
Therefore, there are 72 permutations of the letters in the word "TEXAS" that begin with 2 consonants.