Find the number of arrangements of the letters of the word STRENGTH if the
order of the consonants must not be changed?
Since there are 6 consonants, the E can be in any of seven positions.
I see 8 positons, which include both before and after all the consonants and the present one.
My bad. There are seven consonants.
Guess I need to go back to counting school.
To find the number of arrangements of the letters of the word "STRENGTH" while keeping the consonants in the same order, we can follow these steps:
Step 1: Count the number of consonants in the word "STRENGTH." In this case, the consonants are S, T, R, N, and G, which means there are 5 consonants.
Step 2: Find the number of ways to arrange the vowels (E) in the remaining positions. In the word "STRENGTH," there are 3 vowels (E). We need to find the number of arrangements for these 3 vowels. Since the order of the consonants must not be changed, we can consider the vowels as indistinguishable from each other. In other words, if we swap two E's, the arrangement remains the same. Therefore, we can use the concept of combinations to find the number of arrangements. We can select 3 positions for the vowels out of 8 available positions. The number of ways to do this can be calculated using the combination formula, which is denoted as nCr. In this case, n is the total number of positions (8) and r is the number of positions to fill with the vowels (3). Hence, the number of arrangements for the vowels is 8C3.
Step 3: Multiply the number of arrangements of the consonants (which is 1) by the number of arrangements of the vowels. This will give us the total number of arrangements while keeping the consonants in the same order.
Therefore, the number of arrangements of the letters in the word "STRENGTH" while keeping the consonants in the same order can be calculated as 1 multiplied by 8C3.
Let's calculate 8C3:
8C3 = (8!)/(3!(8-3)!)
= (8!)/(3!5!)
= (8 * 7 * 6 * 5!)/(3 * 2 * 1 * 5!)
= (8 * 7 * 6)/(3 * 2 * 1)
= 56
Therefore, the total number of arrangements of the word "STRENGTH" while keeping the consonants in the same order is 1 * 56, which is equal to 56.