How many arrangements of the word BEGIN are there which starts with a vowel?
no
XYYYY , where X is the vowel, Y can be anything else
number of ways = (2)(4!) = 48
e.g.
E(BGIN) - one of the 24 ways to arrange BGIN
I(GBEN) - one of the 24 ways to arrange GBEN
To find the number of arrangements of the word BEGIN which start with a vowel, we need to consider the words that start with the vowels E or I.
Step 1: Determine the total number of arrangements of the word BEGIN.
The word BEGIN is 5 letters long, so it has 5! (5 factorial) arrangements.
5! = 5 x 4 x 3 x 2 x 1 = 120
Step 2: Determine the number of arrangements that start with the vowel E.
To find this, we fix the vowel E in the first position and permute the remaining 4 letters (B, G, I, N). So we have 4! arrangements.
4! = 4 x 3 x 2 x 1 = 24
Step 3: Determine the number of arrangements that start with the vowel I.
To find this, we fix the vowel I in the first position and permute the remaining 4 letters (B, G, E, N). So we have 4! arrangements.
4! = 4 x 3 x 2 x 1 = 24
Step 4: Add the number of arrangements that start with the vowels E and I.
24 + 24 = 48
Therefore, there are 48 arrangements of the word BEGIN which start with a vowel.
To find the number of arrangements of the word BEGIN that start with a vowel, we can break down the problem into smaller steps:
Step 1: Count the number of total arrangements of the word BEGIN.
The word BEGIN has 5 letters. The number of total arrangements can be calculated using the formula for permutations of distinct items, which is n! (n factorial). In this case, n = 5, so the total number of arrangements of the word BEGIN is 5!.
Step 2: Count the number of arrangements where the word starts with a vowel.
In this case, we want to find the number of arrangements where the first letter is a vowel. The possible vowels at the beginning are "E" and "I". Since the remaining letters can be arranged in any order, we need to consider the number of arrangements for the remaining 4 letters (B, E, G, N). This can be calculated using 4!.
Step 3: Calculate the number of arrangements that satisfy the condition.
To determine the number of arrangements of the word BEGIN that start with a vowel, we need to subtract the number of arrangements in Step 2 from the total number of arrangements in Step 1.
So, the number of arrangements of the word BEGIN that start with a vowel is 5! - 4!.
Simplifying the equation:
5! = 5 x 4 x 3 x 2 x 1 = 120
4! = 4 x 3 x 2 x 1 = 24
The number of arrangements that start with a vowel = 120 - 24 = 96.
Therefore, there are 96 arrangements of the word BEGIN that start with a vowel.